Equation of a Line

How do you make sure that you have the best doctors and the best conditions for patients?  First there was a push for doctors to get more sleep.  Now there is a push to make sure that doctors are getting more hours to finish their work.  Take a look at the debate in this current event.  Write down the broad themes in this article, and the specific details that will make you sound informed.  Then try linking this current event to the following previous SAT essay prompts:  Is there always another explanation or another point of view?  Can success be disastrous?  Should people let their feelings guide them when they make important decisions?  Should people change their decisions when circumstances change, or is it best for them to stick with their original decisions?

Geometry: Coordinate Geometry

Read the following SAT test question and then select the correct answer.

Always read the question carefully and identify the bottom line.  Assess your options for reaching the bottom line, and use the most efficient method to attack the problem.  When you have an answer, loop back to verify that your answer matches the bottom line.

If the graph of the function f is a line with slope 2, which of the following could be the equation of f?

Bottom Line: WOTF (which of the following)

Assess your Options:  For a “which of the following” question you should look at the answers choices, but not until you have used what you know about the equation of a line to decide what kind of equation you need to find.  Start with the information that you are given.

Attack the Problem:  Remember the generic equation for a line is y = mx + b.  In any equation, f(x) and y can mean the same thing.  The variable m is the slope of the line.  You know that your slope must be 2.  Plug that 2 into the equation.  You now have:

f(x) = 2x + b

(The variable b is the y-intercept.  You were not told anything about the y-intercept, so that could be any number.  All you need to do is match the part that you do know, the 2x.)

Loop Back:  You used all the information that you were given, so look down at your answer choices.

(A) f(x) = 4x - 2
(B) f(x) = 2x + 4
(C) f(x) = -2x – 2
(D) $f(x)=\frac{1}{2}x+2$
(E) $f(x)=-\frac{1}{2}x+\frac{1}{2}$

On sat.collegeboard.org, 64% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Circles

Anytime something new happens to something very old, the result is a rich current event that could be interpreted in many different ways.  The Catholic Church has chosen a new pope, and for the first time ever, the pope is from the Americas.  Look for broad themes in this article that would make it easy to relate this current event to an SAT essay topic.

Geometry: Circles

Read the following SAT test question and then select the correct answer.

Read the question carefully and identify the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that your answer matches the bottom line.
In the figure above, a shaded circle with diameter  is tangent to a large semicircle with diameter  at points C and D. Point is the center of the semicircle, and  is perpendicular to . If the area of the large semicircle is 24, what is the area of the shaded circle?

Bottom Line: A sm =? (What is the area of the small, shaded circle?)

Assess your Options:  There are two good ways to approach this problem.  Both ways require you to know the formula for the area of a circle. On collegeboard.org you will find a method that is especially efficient for students who are good at writing equations.  The method used here will focus on geometry skills and estimation in order to avoid the mistakes that often come with working more abstract formulas.

Attack the Problem:  You know the most about the large circle, so start there.  A semicircle is just half of a whole circle.  Therefore, to find the area of the whole circle, you would simply double the 24.

24 × 2 = 48

If you know the area of the large circle, you can use the area formula to find out more information.  The area of a circle is $A=\Pi(r)^{2}$  Plug in the area you just found to find the radius.

Note: working backwards using the area formula for a circle is difficult, because using pi will always result in icky decimals.  If you glance at your answer choices, all of them are whole numbers.  You can estimate pi as 3 instead of 3.14 in order to keep this problem as easy as possible.

48 = 3r²
16 =
4 = r

You now have the radius for the big circle.  Now look back up at the diagram.  The radius for the big circle is also the diameter for the little circle!  If the diameter of the little circle is 4, the radius will be half of that.  Once you know that the radius of the little circle is 2, you are ready to find the area!

$A=\Pi(r)^{2}$
A = 3 × 2²

A = 3 × 4
A = 12

Loop Back:  You found the area of the small circle, so you are ready to look at your answer choices.

(A) 8
(B) 10
(C) 12
(D) 14
(E) It cannot be determined from the information given.

On sat.collegeboard.org, 56% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Coordinate Geometry

Geometry: Coordinate Geometry

Approach every question the same way to minimize mistakes.  Start by reading the question carefully and identifying the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that it matches your bottom line.

In the xy-plane, line l passes through the points (0, 0) and (2, 5). Line m is perpendicular to line l. What is the slope of line m?

Bottom line: slope m = ?

Assess your Options:  You could draw out a graph and solve this visually, but that is a waste of time if you know the formula to find the slope of a line.

Attack the Problem:  You are given the most information about line l, so start with that line.  You should have the formula for slope memorized:

$slope=\frac{rise}{run}=\frac{\Delta y}{\Delta x}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

It is easiest just to think about slope as the change in y-values over the change in x-values.  If you look up at the original points that you have been given, from zero the y-values go up to 5 and the x-values go up to 2.  You now have 5 over 2.

The slope of line l is $\frac{5}{2}$.

At this point, some students will think they are finished and select answer (D).  However, your bottom line was the slope of line m!  The problem tells you that line m is perpendicular to line l.  In order to find a perpendicular line, you must take the opposite reciprocal of the first line; in essence you must flip the sign (negative or positive) and the numbers (a fraction or whole number).

The slope of line m is $-\frac{2}{5}$.

(A) $-\frac{5}{2}$
(B) $-\frac{2}{5}$
(C) $\frac{2}{5}$
(D) $\frac{5}{2}$
(E) 5

On sat.collegeboard.org, 47% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Coordinate Geometry

Geometry: Coordinate Geometry

Read the following SAT test question and then select the correct answer.

Read the question carefully and identify the bottom line.  Assess your options for solving the problem and use the most efficient method to attack the problem.  When you have an answer, loop back to verify that it matches the bottom line.

What is the area of the triangle in the figure above?

Bottom Line: a =?  (What is the area?)

Assess your Options:  The best way to solve this problem is to use the formula for the area of a triangle.  You have already been given all the information that you need to solve the problem.

Attack the Problem:  Start with the formula for the area of a triangle.

$area =\frac{1}{2}(base)(height)$

The base of the triangle extends to the right of the origin (5 units).  The height of the triangle extends upwards from the origin (3 units).

$area =\frac{1}{2}(5)(3)$

Work with the easy numbers first: 5 times 3 is 15.  If you divide 15 by 2 you get 7.5.

Loop Back:  You solved for area, so you are ready to look down at the answer choices.

(A) 4.0
(B) 7.5
(C) 8.0
(D) 8.5
(E) 15.0

On sat.collegeboard.org, 75% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Coordinate Geometry

Isn't it fascinating that no matter how long people study people, there is still more to learn?  Take a look at this current event article that endeavors to explain why women talk more than men.  Pick out the broad topics in this article.  How could you use the facts from this article to support a position on the following SAT essay prompts?

1. Do we need other people in order to understand ourselves?
2. Should heroes be defined as people who say what they think when we ourselves lack the courage to say it?
3. Are people best defined by what they do?

Geometry: Coordinate Geometry

Read the following SAT test question and then select the correct answer.

Always read the question carefully and identify the bottom line.  Assess your options for reaching the bottom line and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that the answer matches the bottom line.

What is the equation of the line parallel to the x-axis and four units above the x-axis?

Bottom Line: equation of a line

Assess your Options:  You could look down at the answer choices, but if you look down without thinking first you will often confuse yourself.  Instead, use the information that you are given to write an equation.

Attack the Problem:  You know that you are dealing with an x-axis, which means you must use a normal xy-graph with a vertical y-axis and a horizontal x-axis.  Draw this on your paper.  Next, imagine 4 ticks on the y-axis and put a little dot four units above the x-axis.  Draw a horizontal line that is parallel to the x-axis.  Does that line ever leave y = 4?  No!  That is the equation of the line.

Note:  If you write x = 4, you create a vertical line.  Think about it this way: the x values change from negative infinity to positive infinity.  If you choose a single x value, the line along this value cannot be parallel to the x-axis because it is limited to a single value.

Loop Back:  You needed an equation of a line, and not necessarily one that mentioned x at all.  You found one.  Look down at your answer choices.

(A) x = -4
(B) x = 4
(C) y = -4
(D) y = 0
(E) y = 4

On sat.collegeboard.org, 67% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Coordinate Geometry

Geometry: Coordinate Geometry

Read the following SAT test question and then select the correct answer.

Always be sure to read the question carefully and make a note of the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that it matches the bottom line.

In the xy-plane, line l passes through the points (a, 0) and (0, 2a), where a > 1.  What is the slope of line l?

Bottom Line: slope of l = ?

Assess Your Options: You could select a number larger than 1, plug it in for the variable a, and then work the problem.  However, if you peek down at the answer choices, notice that some have a variable still in the problem.  It will take you longer to plug in a number than to work the problem using the variables.

Attack the Problem: Your bottom line is a slope, so use the formula for the slope of a line. The formula for slope of a line is:

$\frac{rise}{run}\: or\: \frac{\Delta y}{\Delta x}$

To find the change in y coordinates, subtract the first y-value from the second y-value.  Do the same with the x values:

$\frac{2a-0}{0-a}=\frac{2a}{-a}=-2$

The variable will cancel when you simplify the problem.  Your answer is -2.

Loop Back:  You found the slope of the line, so you are ready to look down at your answer choices.

(A) -2
(B)$-\frac{1}{2}$
(C) 2
(D) -2a
(E) 2a

On sat.collegeboard.org, 42% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Triangles

Geometry: Triangles

Read the following SAT test question and then select the correct answer.

Always read the question carefully so that you don’t misapply any information.  Identify the bottom line and assess your options for solving the problem.  Choose the most efficient method to attack the problem.  When you think you have the answer, loop back to make sure that it matches the bottom line.

If triangle ABC above is congruent to triangle DEF (not shown), which of the following must be the length of one side of triangle DEF?

Bottom Line: side of DEF = ?

Assess your Options: Many students go straight to the Pythagorean Theorem whenever they see a right triangle.  This formula, a² + b² = c², will not help you in this case because you do not know a or b.  Instead, use your knowledge of special triangles to solve this problem.

Attack the problem:  As soon as you see that this is a 30° – 60° – 90° triangle, you should think about the sides that relate to this special triangle.  Those sides, which you should have memorized, are x - x√3 – 2x.  Remember that the longest side has to be across from the biggest angle, the 90° angle.  That is your 2x.  This triangle has a 12 in that position.  Solve for x.

2x = 12
x = 6

Now you know that the side across from the 30° angle, AB, must be 6.  Label it.  Look at the side across from the 60° angle.  AC must be x√3.  You know that x = 6, so this side must be 6√3.  Label it.  You now know all the sides of triangle ABC:

x - x√3 – 2x
6 -6√3 – 12

Your bottom line is a side on triangle DEF, not on triangle ABC.  However, the problem tells you that ABC is congruent to DEF.  Congruent triangles have the same shape and size; they are basically the same triangle with different labels.  That means that the side lengths from triangle DEF will match the lengths you already found for ABC.

Loop back: You took into account all of the information that you were given and solved for your bottom line.  Look down at your answer choices.  One of the three side lengths you found will be there.

(A) 18
(B) 24
(C) 3√6
(D) 6√3
(E) It cannot be determined from the information given.

On sat.collegeboard.org, 38% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Multiple Figures

Geometry: Multiple Figures

Read the following SAT test question and then select the correct answer.

Use the same steps for every math problem.  First, read the question carefully and identify the bottom line.  Next, assess your options and choose the most efficient method to attack the problem.  Finally, loop back to verify that your answer addresses the bottom line.

In the figure above, if PQRS is a quadrilateral and TUV is a triangle, what is the sum of the degree measures of the marked angles?

Bottom Line:  Sum of degrees of the marked angles = ? (Write Sd = ?)

Assess your Options:  You could try to find the individual angles, but you don’t have enough information to do this.  Instead, use the rules you have memorized about each shape.

Attack the Problem:  You know that TUV is a triangle.  All the angles of a triangle add up to 180 degrees.  You know that PQRS is a quadrilateral.  All the angles of a quadrilateral add up to 360 degrees.  In the image, you can see that all of these angles in each of these two shapes are marked, and you know that you are looking for a sum, so add them together.  180 + 360 = 540.

Loop back: Your answer is in degrees and you have found the total of all the marked angles.  Look down at your answer choices.

(A) 420
(B) 490
(C) 540
(D) 560
(E) 580

On sat.collegeboard.org, 77% of the responses were correct.

For more help with SAT writing, visit www.myknowsys.com!

Lines

Geometry: Lines and Angles

Read the following SAT test question and then select the correct answer.

Work all math problems the same way.  Read the problem carefully, identify the bottom line, and assess your options for solving the question.  Choose the most efficient method to attack the problem, and loop back to make sure that your answer matches the bottom line.

Four distinct lines lie in a plane, and exactly two of them are parallel. Which of the following could be the number of points where at least two of the lines intersect?

I. Three
II. Four
III. Five

Bottom Line: # intersections

Assess your Options: You could just start drawing any combination that you can think of, but try to think of the particular answer choices that you are given.  Examine options I, II, and III independently.

Attack the Problem: Think first about option I.  Go ahead and draw out two horizontal parallel lines.  The other lines cannot be parallel to these lines or to each other because the problem says that there are "exactly" two parallel lines.  How could you create three intersections?  One way is to make those next two lines into an “X” and put the middle of the X on one of the preexisting parallel lines.  If you extend the legs of the X out far enough (remember these are lines, not line segments), they will cross the other parallel line in two places.  You have created an image with 3 intersections while following all of the stipulations.

Now turn your attention to option II.  If you have two parallel lines, those lines will never cross.  You know that the other two lines cannot be parallel because this situation has "exactly two" parallel lines.  If you have two lines that are not parallel, no matter how close their slopes are, eventually they must cross.  That is why representing these lines as an X is a good idea.  Any combination of the lines other than the above combination will result in 1 intersection between the lines that are not parallel and 4 intersections where these lines cross the parallel lines for a total of 5 intersections.  There is no way to get only 4 intersections.

Look at option III.  You already thought about it conceptually while examining option II, but you can prove this possibility by drawing a picture.  Draw two horizontal parallel lines.  You can keep the other two lines as an X, but move the middle of the X off of the parallel lines.  Whether the middle of the X is between the parallel lines, above them, or below them, you will now have 5 intersections while following all of the stipulations. (This also confirms your thinking about option II.)

Loop Back:  You examined each option separately, so you are ready to look down at your answer choices.

(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) I, II, and III

On sat.collegeboard.org, 29% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Triangles

Geometry: Triangles

Read the following SAT test question and then select the correct answer.

Approach all math questions the same way so that you can be confident in your method.  Start by reading the question carefully and making a note of the bottom line – the answer that you must find.  Then, assess your options and choose the most efficient method to attack the problem.  When you have found an answer, loop back to make sure that it is your bottom line.

In the triangles above, 3(y – x) =

Bottom Line: 3(yx) = ?  (Don’t solve for x or y and think that you are finished!)

Assess your Options:  The wonderful thing about geometry questions is that there is often more than one way to get to an answer.  The tricky thing is that using some geometry rules will take longer than others.  For example, you could use the rule that all degrees in a triangle add up to 180 degrees.  Then you would write out an equation to solve for the missing variables in each triangle.  This is the method used on collegeboard.org.  However, if you have special triangles memorized, you can save a lot of time.

Attack the Problem:  The first triangle is a right isosceles triangle.  You know this because it has one right angle, and the other two angles are equal.  This is a special triangle that is very common, so you should memorize the fact that its angles measure 45, 45, and 90 degrees.  The x is equal to 45.

Now look at the second triangle.  It is an equilateral triangle.  You know this because all three angles are equal.  You should memorize the fact that all the angles in an equilateral triangle equal 60 degrees.  The y is equal to 60.

Now that you know what the x and y are, plug these numbers into your equation.

3(yx) =
3(60 – 45) = 45

Loop Back:  You solved for your bottom line, so you are ready to look at the answer choices.

(A) 15
(B) 30
(C) 45
(D) 60
(E) 105

On sat.collegeboard.org, 77% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Triangles

Geometry: Triangles

Read the following SAT test question and then select the correct answer.

You should start by reading the problem carefully and identifying your bottom line.  Then assess your options and choose the most efficient method to attack the problem.  Finally, loop back to make sure that the answer you found matches the bottom line that you set out to find.

Which of the following CANNOT be the lengths of the sides of a triangle?

Bottom Line: The word “CANNOT” tells you that you are looking for something that is not true.

Assess your Options:  Problems that ask you to find what cannot be true are often impossible to predict; you will have to look at the answer choices to determine whether they work.  Go ahead and look at the answers.

(A) 1, 1, 1
(B) 1, 2, 4
(C) 1, 75, 75
(D) 2, 3, 4
(E) 5, 6, 8

You could try to use logic and your experience with triangles to eliminate some choices.  For example, answer choice A is not the answer because you know there is such a thing as an equilateral triangle and the sides 1, 1, and 1 would create that kind of triangle.  However, after that point you would probably just be guessing.  If you know the Triangle Inequality Theorem, you can systematically check each answer choice.

Attack the problem:  When you are dealing with three sides of a triangle and you do not know that the triangle is a right triangle, you should always think of the Triangle Inequality Theorem.  This theorem states that for any triangle, side x is less than the sum and greater than the difference of the other two sides.  In other words, each side of the triangle must be less than the other two sides added together and greater than the difference of the other two sides.  If you have a triangle with sides x, y and z, you would write the theorem this way:
|yz| < x < y + z. For the subtraction part you can use absolute value or just always do the bigger side minus the smaller side; the result will be the same. The easiest way to think about the theorem is this: for any triangle, other sides subtracted < one side < other sides added.

Start by checking (E).  It has sides 5, 6, and 8.  Plug these sides into your formula by using the first side, 5, as your x.  Make sure that 5 is greater than the difference of the other two sides, but smaller than the other two sides added together.

|y – z| < x < y + z
8 - 6 < 5 < 6 + 8
2 < 5 < 14
This is true!  (E) works as a triangle.

Now check (D). It has sides 2, 3, and 4.  Plug those numbers into the Inequality Theorem.
|y – z| < x < y + z
4 – 3 < 2 < 3 + 4
1 < 2 < 7
This is true!  (D) works as a triangle.

Now check (C).  It has sides 1, 75, and 75.  Plug those numbers into the Inequality Theorem.
|y – z| < x < y + z
75 – 75 < 1 < 75 + 75
0 < 1  < 150
This is true!  (C) works as a triangle.

Now check (B).  It has sides 1, 2, and 4.  Plug those numbers into the Inequality Theorem.
|y – z| < x < y + z
4 – 2 < 1 < 2 + 4
2 < 1 < 6.
Is 2 less than 1?  No!  This is false.  You cannot have a triangle with these three side lengths.

Loop Back:  Your bottom line was to find an answer choice that cannot be a triangle, so you are finished!

On sat.collegeboard.org, 49% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Read the following SAT test question and then select the correct answer.

Math questions should always be read carefully.  You will also avoid making errors by identifying the bottom line and assessing your options for solving the question.  Choose the most efficient method to attack the problem.  When you have finished, loop back to be sure that even if there were multiple steps, you reached the bottom line.

The length of a rectangle is increased by 20%, and the width of the rectangle is increased by 30%. By what percentage will the area of the rectangle be increased?

Bottom Line: % change = ?

Assess your Options:  You could work this problem without picking any numbers; however, picking easy numbers will allow you to think about the problem in a more concrete way and avoid errors.

Attack the Problem:  One of the easiest numbers to work with is one.  Think of your original rectangle as having a length of one and a width of one.  The formula for area of a rectangle is length times width.  If L × W = A, for your first rectangle you have 1 × 1 = 1.   The area of the original rectangle is one.

Then think about the changes that occur to that rectangle.  The length increases by 20%.  In order to find 20% of 1, all you have to do is move the decimal over twice to .2.  The new length is 1.2.  Use the same method to find the new width, and an increase of 30% becomes 1.3.  The area of the rectangle after the change is 1.2 × 1.3 = 1.56

The formula for percent change would require you to find the difference between these two areas and divide that by the original number.  You use the same formula whether you are looking for an increase or a decrease.  Notice that your original number is one, so dividing by one will not change your answer.  All you need to do is find the difference between the areas: 1.56 – 1 = .56.  What is .56 as a percent?  Your answer is 56%.

Loop Back:  You found the percent change, which was your bottom line.  Look down at your answer choices.

(A) 25%
(B) 36%
(C) 50%
(D) 56%
(E) 60%

On sat.collegeboard.org, 34% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Parabolas

Read the following SAT test question and then select the correct answer.

Don’t let this question intimidate you just because it has a parabola.  Use the same method that you would use with any other math problem.  Read the question carefully, identify the bottom line, and choose an efficient method to solve the problem.  Then attack the problem and loop back to make sure that you solved for the bottom line.

The quadratic function f is graphed in the xy-plane above. If f(x) ≤ u for all values of x, which of the following could be the coordinates of point P?

Your bottom line is which values could be the coordinates of point P, so make a note of the bottom line on your paper, and start with what you know about this point.  You are told that f(x) ≤ u for all the values of x.  That is your y value, so that is just letting you know that nothing can be higher than u, which is on point P.  If you are looking for the highest point on a downward opening parabola, what are you actually looking for?  The vertex!

Think about it this way: as the parabola extends outward from the vertex, both sides stay an equal distance from the vertex. You have just examined the information given about the y-axis, so turn your attention to the x-axis.  You are given two x values that are of equal height on your parabola, so the x value of the vertex, P, must be exactly between them.  Your highest value is 4, so you might be tempted to halve 4 and get 2.  Just be sure to remember that the first point is not at zero, but at 1.  That means that your parabola has been shifted 1 unit to the right.  To find the midpoint, use the midpoint formula, which is simply an average of the two numbers that you have.

$\frac{1 +4}{2}=2.5$

You now have the x value of 2.5. You are not given any additional information about the limits of the y-axis, so loop back to the bottom line.  The question is not actually asking you to find both x and y coordinates.  Remember that your bottom line is what “could” be the coordinates, so this is probably enough information to find the correct answer.  Look down at your answer choices now.

(A) (2, 3.5)
(B) (2.25, 3.25)
(C) (2.5, 3)
(D) (2.75, 4)
(E) (3, 2.5)

All you know about the y value is that it must be greater than 0, so all of the y values will work, but only one of the answers has the x value of 2.5.

On sat.collegeboard.org, 53% of the responses were correct.

For more help with math, visit

Circles

Would you like a break from school right around now?  Students in Chicago are getting just that.  They have had over a week without school as teachers strike over conflicts with the mayor concerning the best methods to improve underperforming schools.  Take a look at this current event and think about how you could use details from this story in an excellent example for your essay that shows SAT graders that you are well informed.  Make a list of all of the broad themes that this story illustrates, such as change, education, other points of view, adversity, success (Can it be disastrous?), how to question those in authority, and many many more.

9/18 Circles

Read the following SAT test question and then select the correct answer.

The circle above has center P. Given segments of the following lengths, which is the length of the longest one that can be placed entirely inside this circle?

Your bottom line is the longest length that can be placed inside a circle.  Before you look at any of the numbers in this particular problem, think about circles in general.  You know that a line segment, within the circle, with both end points on the circle is called a chord.  The longest chord will always be the diameter of the circle.

You have enough information to find the diameter of this circle.  The line labeled 4 is actually the radius of the circle.  A radius is half of the diameter, so multiply the radius by 2 in order to find the diameter: 2 × 4 = 8.

This is the part of the problem where you loop back and see whether your answer matches the bottom line.  You found the diameter of the circle, but that is not actually the bottom line.  A diameter is a chord that touches the circle’s edges.  You were asked to find the longest length that can be placed entirely in the circle.  In order for the length you found to not touch the circle, it must be slightly smaller than 8, perhaps 7.9999999.  Now look down at your answer choices.

(A) 6.99
(B) 7.00
(C) 7.99
(D) 8.10
(E) 14.00

Notice that you did not need to use all of the information that you were given in this problem.  Always take the time to read the question carefully so that you will not be confused or distracted by extra information, such as the chord labeled 7 in this problem.

On sat.collegeboard.org, 62% of the responses were correct.

For more help with writing, visit www.myknowsys.com!

Multiple Figures

SAT geometry questions mention basic shapes such as squares and cubes or circles and spheres that are all around us in the natural world.  One sphere that people have always looked towards at night is the Moon.  Right now, people around the world are remembering the life of Neil Armstrong, the first man to set foot on the Moon.  Neil Armstrong is an excellent historical figure to mention in your SAT essay.  Review a few facts about the life of this famous man here.  See how Americans are responding to his death here.

8/28 Geometry: Multiple Figures

Read the following SAT test question and then select the correct answer.

Geometry questions often require you to add labels to a diagram, so you must be especially careful to note exactly which information you are given when you read the question.  As always, make a note of the bottom line, assess your options for efficiently solving the problem, attack the problem, and loop back to make sure that you have answered the bottom line.  Writing what you know neatly will often help you see new ways to work with the shapes you are given.

In the figure above, O is the center of the circle and  is equilateral. If the sides of  are of length 6, what is the length of ?

Geometry problems can be difficult if you are not sure how to attack the problem.  Think of these kinds of problems as puzzles; use the pieces of information and the rules that come to your mind.  There are multiple ways of arriving at the correct answer, but this is one of the fastest ways to get there.

The first information that you are given is about an equilateral triangle (Triangle ABO).  Identify the equilateral triangle and label all of the interior angles 60̊°.  All equilateral triangles only have angles of 60°.  You are also given the information that the sides of this triangle have a length of 6.  Label all the sides of this triangle as well.

Now look at the information a little differently.  The two triangles inscribed on the circle form a single larger triangle.  You labeled the length of one side as 6 (Side AB).  Look at Side AC.  Line AO forms the radius of the circle, as does Line OC, so both must be the same length.  Your total length of Side AC must be 12.

Here is a rule you should memorize: any triangle that has the diameter of a circle as one of its sides will be a right triangle.  The diameter forms the hypotenuse, so the opposite angle (in this case Angle B) must be 90°.  Once you know two sides of any right triangle, you can find the third.  Before you pull out the Pythagorean Theorem, notice that Triangle ABC is a special triangle.  Angle A is 60° and Angle B is 90°, so Angle C must be 30°.  For any 30-60-90 triangle, the corresponding sides will be x, x√3, and 2x.  In this case, your x = 6 and your 2x = 12, so what is the missing side?  Label the missing side 6√3 and look up at the question to see whether you have found your bottom line.  Then match your answer to the answer choices.

(A) 3√3
(B) 4√3
(C) 6√3
(D) 9
(E) 12

On sat.collegeboard.org, 53% of the responses were correct.

For more help with math, visit www.myknowsys.com!

Solids

Read the following SAT test question and then select the correct answer.

The Knowsys method requires you to read each math question carefully and identify the bottom line.  You must also assess your options to find the best way to attack the problem, solve it, and loop back to make sure that you solved for the bottom line.

A right circular cylinder has height 6 and volume 54π. What is the circumference of its base?

Make a note of the fact that you are solving for the circumference of the base by writing C  = ? under the problem.  You should know that C = 2πr, and based on that you should realize that this problem will require more than one step.  You cannot solve for the circumference of the base without knowing the radius of the base circle.  You do know the volume of the cylinder, so you can use that information to find out more about the base circle.  You should have the formula for the volume of a cylinder memorized: Volume = πr²h. Plug in the values you already know to solve for the radius.

V = πr²h
54π = πr²6   (divide each side by 6π)
9 = r²    (take the square root of both sides)
3 = r

You now have the information that you need for the circumference formula.  Be careful not to look down at your answers yet, because even though you solved part of the problem, you have not yet found the bottom line.

C = 2πr
C = 2π3
C = 6π

(A) 2π
(B) 3π
(C) 6π
(D) 9π
(E) 18π

On sat.collegeboard.org, 41% of the responses were correct.

For more help, visit www.myknowsys.com!

Lines and Angles

Geometry: Lines and Angles

Read the following SAT test question and then select the correct answer.

For every math problem, you should use the Knowsys method: read the question carefully, identify the bottom line, assess your options, attack the problem, and loop back to verify that the answer you found addresses the bottom line.

In the figure above, x = 60 and y = 40. If the dashed lines bisect the angles with measures of x° and y°, what is the value of z?

Geometry questions often include figures with multiple variables.  When you are assessing your options, realize that you can estimate values with figures that are drawn to scale, but that figures that are not drawn to scale may be misleading and estimation may result in a wrong answer.  When you are prepared to attack your problem, it is especially important to write your scratch work so that you can see how each number you find relates to the figure.  The easiest way to do that is to add the values you find to the figure.

The bottom line that you are solving for is z, but the information you are given is about x and y. First look at x.  Your ability to solve this problem hinges on your knowledge that “bisect” means “divides in half.” You know that x totals 60, so half of 60 is on each side of the dashed line that bisects x

60 ÷ 2 = 30

Likewise, you know that y totals 40, so half of 40 is on each side of the dashed line that bisects y.

40 ÷ 2 = 20

Now look at z. This variable overlaps half of x and half of y.  You just solved for each of these, so add them together.

30 + 20 = 50

Loop back to make sure that you solved the question that was asked and then match your answer choice to the answers that are given.

(A) 25
(B) 35
(C) 40
(D) 45
(E) 50

On sat.collegeboard.org, 81% of responses were correct.

For more help with math, visit www.myknowsys.com

Circumference of a Circle

On this day in 1964 President Johnson signed the Civil Rights Act. Surprisingly, it was met by much opposition from both white and African Americans. Many historians now believe it was a major influence in shaping America's social and political development. You can learn more about the Civil Rights Act here.

Geometry: Circumference of a Circle

Remember to read the question carefully. Some students panic when they see a complicated diagram. Every problem on the SAT has a solution that you can reach without any particular, specialized knowledge (though you do still need to memorize basic math formulas). Slow down and reread the problem carefully; make sure that you understand what the question is actually asking.

In the figure above, inscribed triangle  is equilateral. If the radius of the circle is , then the length of arc  is

At first, when you look at this diagram it looks quite complicated. You might know some facts about triangles inscribed in circles, but those facts won't help you in this problem. Instead, remember that after you read the problem carefully, you need to identify the "bottom line." You are looking for the length of arc AXB. Note that there is no label that says "figure not drawn to scale." That means that the figure is drawn to scale (in other words, you could make an estimate based on how the figure looks). It does look like the arc AXB is just 1/3 of the circumference of the circle. In fact, if you think about it, it must be (since the triangle is an equilateral triangle). Since you know that the radius of the circle is r, the diameter must be

$2\pi r$
and therefore, the length of arc AXB is just

$\frac{2\pi r}{3}$

Now, take a look at the answers and select the choice that matches your prediction. Don't forget to loop back and verify that your answer matches the "bottom line."

(A)
(B)
(C)
(D)
(E)

The correct answer choice is (A).

On sat.collegeboard.org 54% of the responses were correct.

For more help with math, visit www.myknowsys.com.

Area of a Triangle

Once again we return to History.com's "This Day in History". On June 2nd, 1935 Babe Ruth retired from baseball. The "Sultan of Swat" set numerous records and led the Yankees to 4 World Series victories. His carrier slugging average of .690 is still the highest in Major League history. Babe Ruth could make an Excellent Example illustrating the importance of dedication to the things one is passionate about.

6/2 Area of a Triangle

What is the area of the triangle in the figure above?

As always, the first step is to read the problem carefully and identify the bottom line. We are looking for the area of the triangle above and we can use the fact that it has been drawn on the coordinate plane to figure out the base and the height. Remember that since there is nothing on the drawing that says "figure not drawn to scale", the figure must be drawn to scale. That means that we could "eyeball" it if we needed to. In this case we can see that the triangles height is 3 and the base is 5. All that we need to solve the problem is the formula for the area of a triangle.

$a=\frac{1}{2}b\times h$

We simply plug in our values for h and b to get

$a=\frac{1}{2}(5)\times (3)=\frac{1}{2}(15)=7.5$

We now look at the answer choices below to see which one matches our prediction.

(A)
(B)
(C)
(D)
(E)

The correct answer choice is (B).

On sat.collegeboard.com 70% of the responses were correct.

For more help with math, visit www.myknowsys.com!

Triangles

History.com's This Day in History is a great place to look for interesting historical events that might otherwise be overlooked. Common examples like Martin Luther King Jr. or the Holocaust will not make your essay stand out, but the fact that on May 9th, 1950, L. Ron Hubbard published Dianetics or that in 2001, soccer fans were trampled in Ghana will make your essay stronger.

5/9 Geometry: Triangles

Remember to always follow the Knowsys Method for math problems. The method will save you time and errors not only on the SAT but also in your regular math classes and problems. First, read the question carefully and identify the bottom line. Once you know what the problem is asking, assess your options by asking "What could I do?" "What should I do?" Select the most efficient method, attack the problem, and loop back to make sure that you answered the question correctly.

If triangle ABC above is congruent to triangle DEF (not shown), which of the following must be the length of one side of triangle DEF?

First, at the top of your scratch work, write one side of DEF = ?

Next, assess your options. How can you find the side lengths of a triangle that is not shown? The problem mentions that ABC and DEF are congruent, which means all their side lengths and angle measurements are the same. That means that you can simply change the labels on ABC to DEF. To find the answer, though, you will need to figure out the side lengths. You could try to use the Pythagorean Theorem here, but it would be very difficult. Instead, you should notice that the triangle is one of the special right triangles that you have memorized. You can use that information to find the side lengths.

Triangle DEF is a 30-60-90, which means the side lengths are $x-x\sqrt{3}-2x$. The hypotenuse is 12, so 12 = 2s and 6 = s. The hypotenuse of the triangle is 12, the short leg is 6, and the other leg is $6\sqrt{3}$.

Loop back to the bottom line. You are looking for any side of Triangle DEF, so now that you have all three, you only need to look at the answer choices and find one that matches any of these three numbers.

A) 18

B) 24

C) $3\sqrt{6}$

D) $6\sqrt{3}$

E) It cannot be determined from the information given.