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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 C 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?
is tangent to
a large semicircle with diameter at
points C and D. Point C 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 ^{2} "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
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 = r²
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!
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