# 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!

# 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!

# 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!

# 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!

# Triangles

Yesterday's Question of the Day about Red Cloud piqued my interest, so I decided to look him up for today's Link of the Day. Red Cloud was an amazingly successful war leader of the Lakota Indians, assaulting several United States Army forts along the Bozeman Trail in the 1860's. By the end of the decade, the US agreed not only to abandon its forts in Lakota territory, but also to guarantee Lakota control over a vast land area, including the western half of modern South Dakota and parts of Montana and Wyoming. Unfortunately, Red Cloud's victories did not last, and eventually the white settlers reclaimed and broke apart the Lakota holdings. Red Cloud's tireless efforts to protect his people and his culture would make an outstanding Excellent Example for your essay.

Let's take this a step further: What kind of essay prompt could you answer with the story of Red Cloud? Please respond in the comments!

## 7/26 > Triangles

Whenever you approach a math problem, remember to follow the Knowsys method. Rather than charging in, take a moment to read the problem carefully and identify the bottom line. Consider the best way to approach the problem--what could I do? What should I do? Then attack the problem and, finally, loop back to the top and make sure you answered what the question was actually asking. The last and easiest step is to match your answer to the provided answer choices.

In triangle ABC, the length of side $\overline{BC}$ is 2 and the length of side $\overline{AC}$ is 12. Which of the following could be the length of side $\overline{AB}$?

First, note the bottom line at the top of your scratch work.

$\overline{AB}$ = ?

Next, consider your options. What does the problem tell you? What strategies, formulas, or theorems do you know that could help you solve it? In this case, the problem tells you that you are dealing with a triangle and supplies two side lengths. With so little information, you really only have one tool that can help you: the Triangle Side Lengths Inequality.

The Side Lengths Inequality states that any side of a triangle must be less than the sum and greater than the difference of the other two sides. When you think through it, this actually becomes fairly obvious. If one side were longer than the other two sides put together, the shape could no longer be a triangle. It would fold flat into a line. If one side were too short, it would not be able to "reach" the other sides and the triangle would just be three line segments rather than a closed shape. The Side Lengths Inequality is usually expressed this way:

$\left | y-x \right |

For simplicity's sake, you can rename the sides of the triangle in the problem x, y, and z rather than shuffling As, Bs, and Cs around. (Be sure to note this in your bottom line!) Now that you've chosen the most efficient way to solve the problem, attack it ruthlessly!

First, take the side lengths you are given and plug them into y and z.

$\left | 2-12 \right |

Next, perform some simple arithmetic to solve for x.

$\left | -10 \right |

$10

Now you've narrowed down the range of possible values of x. Loop back to double-check the bottom line. If you remembered to update it earlier, it should look something like this:

x = $\overline{AB}$ = ?

Since you've found the possible values of x, you've also found the possible lengths of side $\overline{AB}$. The last step is to find an answer choice that matches what you found.

(A) 6

(B) 8

(C) 10

(D) 12

(E) 14

Note that the inequality uses "less than" signs, not "less than or equal to" signs. That means that side $\overline{AB}$ cannot equal 10 or 14; it must be 12. The answer is D.

On sat.collegeboard.org, 45% of responses were correct.

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

# Triangles

Did you know that you can only see color in the central part of your vision? The rest of your vision is actually black and white. Your brain remembers the colors of items that you have looked at and fills them in. Physicist turned webcomic writer Randall Munroe has a great diagram explaining some of the little-known facts about your vision. You can view the diagram here.

## 7/11 Triangles

Whenever you are given a diagram, check to see if there is a note that says "figure not drawn to scale." If you don't see a note, that means that the figure is drawn as accurately as possible. In other words, you can look at the diagram and make estimates. On the other hand, if the figure is not drawn to scale, then it is distorted intentionally to trick you. Focus on the facts you know about the figure and not its appearance.

In the figure above, the circle with center  and the circle with center  are tangent at point . If the circles each have radius , and if line  is tangent to the circle with center  at point , what is the value of ?
The diagram above can look a little intimidating. That makes it even more important that you follow the Knowsys method step by step. First, read the problem carefully and identify the bottom line. You are looking for the value of x. Now think about the different ways you could attack the problem. Notice that the figure is drawn to scale (since there is no note that says "figure not drawn to scale"). You could estimate the value of the angle. However, if you look at the answer choices below, they are so close together that an estimation won't really do you much good. Instead, it's best to fill in pieces of the diagram step by step until you can find the value of x. Don't forget to "attack the problem". A positive attitude can make a big difference when you are working a challenging math problem.

You know that the radius of both circles is 10. That means that line AB is 10 and line AC is 20. You also know that line l is tangent to the circle at point B. If you remember your geometry rules for circles, you know that angle ABC is 90 degrees. Now, you need to remember the special triangles. You know that a right triangle with a leg of length x and a hypotenuse of length 2x is a 30-60-90 triangle. That means angle x must be 60 degrees. Don't forget to loop back and verify that your answer matches the bottom line.

(A) 55
(B) 60
(C) 63
(D) 65
(E) It cannot be determined from the information given.

The correct answer choice is B.

On sat.collegeboard.org 51% of the 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.

# 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.

On sat.collegeboard.org, 36% of responses were correct.

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

# Circles and Triangles

## Geometry: Circles and Triangles

When you read, make sure you read carefully so that you don't miss anything important. Write down the bottom line, then assess your options and attack the problem with the most efficient method you know. Finally, loop back to make sure you answered the right question.

The circle shown above has center O and a radius of length 5. If the area of the shaded region is  $20\pi$, what is the value of x?

If this problem seems impossible at first glance, don't panic. It will have several steps, but it is far from impossible. Keep in mind that you don't need to know the entire path to the right answer when you start working, and in this problem that would be incredibly difficult. Just follow the steps of the Knowsys Method.

Before you start, notice that the picture says "not drawn to scale." That means that the test makers deliberately distorted it so it wouldn't help you as much, but you can still get some useful facts out of it. For example, O is both the center of the circle and one corner of the triangle. The fact that it is a right triangle is also likely to prove useful.

First, write down the bottom line.

$x=$

Next, assess your options. When I ask my students what they could do when facing a problem like this, sometimes their answer is, "Cry." You could, if it would make you feel better, but on the test that will cost you time, and during practice it won't make the problem go away. So what do you do next?

Look at what information the problem gives you. You have the radius of the circle and the area of part of the circle. You can use the radius to find the total area...

$a=\pi r^{2}$

$a=\pi 5^{2}$

$a=25\pi$

...and then compare the two amounts.

$\frac{20\pi }{25\pi }$

$\frac{4 }{5 }$

Now you've figured out that the shaded area is four-fifths of the total area of the circle. What can you do with that? Well, the remaining fifth of the circle is within the triangle, which means that the corner with its vertex at O has a fifth of the degrees around the center of the circle.

$360\ast \frac{1}{5}=72$

So that angle measures 72 degrees. Since this is a right angle, it is now easy to calculate x.

$x=180-90-72=18$

Glance up to the bottom line to make sure you solved what you needed to. Then, look at the answer choices and select the right one.

A) 18

B) 36

C) 45

D) 54

E) 72

On sat.collegeboard.org, 33% of responses were correct.

Want more help with math? Visit myknowsys.com!

# Triangles

## Geometry: Triangles

A 25-foot ladder is placed against a vertical wall of a building, with the bottom of the ladder standing on concrete feet from the base of the building. If the top of the ladder slips down feet, then the bottom of the ladder will slide out...

The first step in the Knowsys Method is to read carefully. You should notice a few things:
1. This is a right triangles problem involving the ladder, the ground, and the wall.
2. The problem provides two sides of the triangle that is formed before the ladder slips: 25 ft and 7 ft.
3. The 4 ft is not a measure of the second triangle, but instead a measure of change. It tells how far the ladder slipped. Be careful how you use this number!
After reading carefully, you should look for the Bottom Line. What is this problem actually asking for? In almost all cases, the Bottom Line is found at the very end of the problem text. This one is asking how far the bottom of the ladder slides out. At the top of your scratch work, abbreviate this so you know what you’re looking for.

One more piece of Knowsys advice: Be methodical. At first glance, it looks like the answer to this question is 4; after all, if the top of the ladder slid down four feet wouldn't the other end move the same distance? This is a TRAP! If the question were that easy, it wouldn't be on the SAT. Remember that sometimes SAT questions have counter-intuitive answers, so you should be sure to stick with the math instead of what "feels right."

Next, assess your options. Stop and consider “What could I do?” This means you open up your “mental toolbox” to look for any formulas or strategies that could help you. In this case, the Pythagorean Theorem $a^{2}+b^{2}=c^{2}$ will probably spring to mind. You could use the Pythagorean Theorem to find the missing side of the first triangle, or you could memorize the Pythagorean Triplets—right triangles with whole numbers on all three sides—and recognize the 7-24-25 triangle in this problem. After that, you will need to find out the dimensions of the second triangle and subtract to find out how far the other end of the ladder moved. Now that you've decided what you SHOULD do, attack the problem fearlessly!

First, the 7-24-25 triangle shows that the ladder starts off touching the wall 24 feet above the ground. After it slips down 4 feet, it will touch the wall 20 feet above the ground. The ladder, which is also the hypotenuse, is still 25 feet. The problem has conveniently given you another Pythagorean Triplet, the 15-20-25 triangle; after the ladder slips, the base will be 15 feet away from the wall.

This is when many students are tempted to stop. I finished the triangle! I’m done! Loop back to make sure you answered the correct question. You were looking for how far the ladder moves, so there is one more step. The new distance (15) minus the original distance (7) is 8 feet. That is what the question is looking for. Now check the answer choices:

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