2013 AMC 8 Problems/Problem 23
Contents
Problem
Angle of
is a right angle. The sides of
are the diameters of semicircles as shown. The area of the semicircle on
equals
, and the arc of the semicircle on
has length
. What is the radius of the semicircle on
?
Video Solution for Problems 21-25
Video Solution
https://youtu.be/crR3uNwKjk0 ~savannahsolver
Solution 1
If the semicircle on were a full circle, the area would be
.
, therefore the diameter of the first circle is
.
The arc of the largest semicircle is , so if it were a full circle, the circumference would be
. So the
.
By the Pythagorean theorem, the other side has length , so the radius is
Solution 2
We go as in Solution 1, finding the diameter of the circle on and
. Then, an extended version of the theorem says that the sum of the semicircles on the left is equal to the biggest one, so the area of the largest is
, and the middle one is
, so the radius is
.
Video Solution by OmegaLearn
https://youtu.be/abSgjn4Qs34?t=2584
~ pi_is_3.14
See Also
2013 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
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