2013 AMC 8 Problems/Problem 23: Difference between revisions

Revision as of 20:22, 9 August 2025

Problem

Angle $ABC$ of $\triangle ABC$ is a right angle. The sides of $\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\overline{AB}$ equals $8\pi$ , and the arc of the semicircle on $\overline{AC}$ has length $8.5\pi$ . What is the radius of the semicircle on $\overline{BC}$ ? $[asy] import graph; pair A,B,C; A=(0,8); B=(0,0); C=(15,0); draw((0,8)..(-4,4)..(0,0)--(0,8)); draw((0,0)..(7.5,-7.5)..(15,0)--(0,0)); real theta = aTan(8/15); draw(arc((15/2,4),17/2,-theta,180-theta)); draw((0,8)--(15,0)); dot(A); dot(C); label("$A$", A, NW); label("$B$", B, SW); label("$C$", C, SE);[/asy]$

$\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 7.5 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 8.5 \qquad \textbf{(E)}\ 9$

Video Solution

https://youtu.be/crR3uNwKjk0 ~savannahsolver

Solution 1

If the semicircle on $\overline{AB}$ were a full circle, the area would be $16\pi$ .

$\pi r^2=16 \pi \Rightarrow r^2=16 \Rightarrow r=+4$ , therefore the diameter of the first circle is $8$ .

The arc of the largest semicircle is $8.5 \pi$ , so if it were a full circle, the circumference would be $17 \pi$ . So the $\text{diameter}=17$ .

By the Pythagorean theorem, the other side has length $15$ , so the radius is $\boxed{\textbf{(B)}\ 7.5}$

~Edited by Theraccoon to correct typos.

Brief Explanation

SavannahSolver got a diameter of $17$ because the given arc length of the semicircle was $8.5\pi$ . The arc length of a semicircle can be calculated using the formula $\pi r$ , where $r$ is the radius. let’s use the full circumference formula for a circle, which is $2\pi r$ . Since the semicircle is half of a circle, its arc length is \$pi r $, which was given as$ 8.5\pi $. Solving for$ r $, we get$ r=8.5 $. Therefore, the diameter, which is$ 2r $, is$ 2 $x$ 8.5=17 $Then, the other steps to solve the problem will be the same as mentioned above by SavannahSolver the answer is$ \boxed{\textbf{(B)}\ 7.5}$

. - TheNerdWhoIsNerdy. Minor edits by -Coin1

Solution 2

We go as in Solution 1, finding the diameter of the circle on $\overline{AC}$ and $\overline{AB}$ . 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 $\frac{289\pi}{8}$ , and the middle one is $\frac{289\pi}{8}-\frac{64\pi}{8}=\frac{225\pi}{8}$ , so the radius is $\frac{15}{2}=\boxed{\textbf{(B)}\ 7.5}$ .

~Note by Theraccoon: The person who posted this did not include their name.

Video Solution by OmegaLearn

https://youtu.be/abSgjn4Qs34?t=2584

~ pi_is_3.14

Answer (B) 7.5

~ Mia Wang the Author ~skibidi ~Coin1

@@ Line 40: / Line 40: @@
 <math>8.5\pi</math>. The arc length of a semicircle can be calculated using the formula
 <math>\pi r</math>, where
-r is the radius. let’s use the full circumference formula for a circle, which is
+<math>r</math> is the radius. let’s use the full circumference formula for a circle, which is
-πr. Since the semicircle is half of a circle, its arc length is
+<math>2\pi r</math>. Since the semicircle is half of a circle, its arc length is
-πr, which was given as
+\$pi r<math>, which was given as
-<math>8.5\pi</math>. Solving for
+</math>8.5\pi<math>. Solving for
-<math>r</math>, we get
+</math>r<math>, we get
-r=8.5
+</math>r=8.5<math>
 . Therefore, the diameter, which is
-r, is
+</math>2r<math>, is
-x8.5=17
+</math>2<math> x </math>8.5=17<math>
 Then, the other steps to solve the problem will be the same as mentioned above by SavannahSolver
-the answer is <math>\boxed{\textbf{(B)}\ 7.5}</math>
+the answer is </math>\boxed{\textbf{(B)}\ 7.5}$

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