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Circles

Link of the Day

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.
graphic
In the figure above, a shaded circle with diameter line C D is tangent to a large semicircle with diameter line A B at points C and D. Point is the center of the semicircle, and line A B is perpendicular to line C D. 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   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 = 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.

The correct answer is (C).


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

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

Circles

Link of the Day

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. 

 math image

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

The correct answer is (C).  
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

Link of the Day

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.

graphic

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

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

The correct answer is (C).


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

Now that you have an answer, match your answer to the answer choices.

(A) 2π
(B) 3π
(C) 6π
(D) 9π
(E) 18π

The correct answer is (C).


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

For more help, visit www.myknowsys.com!

Triangles

Link of the Day

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

Read the following question and then select the correct answer.

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.

math image
In the figure above, the circle with center A and the circle with center C are tangent at point D. If the circles each have radius 10, and if line l is tangent to the circle with center A at point B, what is the value of x?
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

Link of the Day

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


Read the following SAT test question and then click on a button to select your answer. 

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.


math image
In the figure above, inscribed triangle A B C is equilateral. If the radius of the circle is r, then the length of arc A X B 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


and therefore, the length of arc AXB is just



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) (2 times pi times r) over 3
(B) (4 times pi times r) over 3
(C) (3 times pi times r) over 2
(D) (pi times r^2) over 3
(E) (2 times pi times r^2) over 3



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.

Circles

Link of the Day

Isaac Newton is one of the most famous scientists of all time, and His book Philosophiae Naturalis Principia Mathematica is arguably the most important book published in the history of science. He was a fascinating and enigmatic character. He suffered several major nervous breakdowns over the course of his life, and he spent many years researching alchemy (this investigation into the hidden forces of nature helped to lead him to his discovery of the action-at-a-distance nature of the law of gravity). You can read more about Newton here. He would make a great "excellent example" for your essay.

6/14 Geometry: Circles


Read the following SAT test question and then click on a button to select your answer. 


graphic

In the figure above, a shaded circle with diameter line C D is tangent to a large semicircle with diameter line A B at points C and D. Point C is the center of the semicircle, and line A B is perpendicular to line C D. If the area of the large semicircle is 24, what is the area of the shaded circle?     



As always, you should start by reading the problem carefully and identifying the bottom line. You are looking for the area of the shaded circle. Recall that you can calculate the area of a circle using the formula 


Now, you can assess your options. In this case, you need to find the value of r (the radius of the shaded circle) to calculate the area of the circle. You are given that the area of the large semicircle is 24, or in other words, the area of the circle (with A-B) as the diameter is 48. At this point you need to be very careful how you proceed. As long as you don't make any algebra errors, there are several ways to solve this problem, but some of them are much easier than others. You know that the area of the semicircle is 24 and you are looking for the area of the smaller circle. You also know that  R (the radius of the semicircle) is twice what r (the radius of the shaded circle) is. Now, you can write the following formulas






Note that all you have above is the area of the semicircle (which is the area of a circle with radius R=2r divided by 2). Since you know that the area of the semicircle is equal to 24 you could solve the formula for r and then calculate the area of the shaded circle. However, notice that you already have the area of the circle calculated above, it's just being multiplied by 2. So instead you can write the following



Remember to loop back and verify that you have answered the bottom line. Recall that at the beginning you defined r as the radius of the shaded circle and R as the radius of the semicircle. Now look at the answer choices and select the one that matches your answer.

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

The correct answer is (C)

on sat.collegeboard.org 56% of the responses were correct.

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

Circles and Triangles

Geometry: Circles and Triangles

Read the following test question and then select your answer.

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.

math image

The circle shown above has center O and a radius of length 5. If the area of the shaded region is  , 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.



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...







...and then compare the two amounts.





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.



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



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


The answer is A.

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



Want more help with math? Visit myknowsys.com!