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Difference between revisions of "2013 AMC 8 Problems/Problem 23"

(→‎Solution 2)
(→‎Answer (B) 7.5)
 
(27 intermediate revisions by 14 users not shown)
Line 3: Line 3:
 
<asy>
 
<asy>
 
import graph;
 
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((0,8)..(-4,4)..(0,0)--(0,8));
 
draw((0,0)..(7.5,-7.5)..(15,0)--(0,0));
 
draw((0,0)..(7.5,-7.5)..(15,0)--(0,0));
Line 8: Line 12:
 
draw(arc((15/2,4),17/2,-theta,180-theta));
 
draw(arc((15/2,4),17/2,-theta,180-theta));
 
draw((0,8)--(15,0));
 
draw((0,8)--(15,0));
−
label("$A$", (0,8), NW);
+
dot(A);
−
label("$B$", (0,0), SW);
+
dot(C);
−
label("$C$", (15,0), SE);
+
label("$A$", A, NW);
−
</asy>
+
label("$B$", B, SW);
 +
label("$C$", C, SE);</asy>
  
 
<math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 7.5 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 8.5 \qquad \textbf{(E)}\ 9</math>
 
<math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 7.5 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 8.5 \qquad \textbf{(E)}\ 9</math>
 +
 +
==Video Solution==
 +
https://youtu.be/crR3uNwKjk0 ~savannahsolver
  
 
==Solution 1==
 
==Solution 1==
−
If the semicircle on AB were a full circle, the area would be 16pi. Therefore the diameter of the first circle is 8. The arc of the largest semicircle would normally have a complete diameter of 17. The Pythagorean theorem says that the other side has length 15, so the radius is <math>\boxed{\textbf{(B)}\ 7.5}</math>.
+
If the semicircle on <math>\overline{AB}</math> were a full circle, the area would be <math>16\pi</math>.  
 +
 
 +
<math>\pi r^2=16 \pi \Rightarrow r^2=16 \Rightarrow r=+4</math>, therefore the diameter of the first circle is <math>8</math>.  
 +
 
 +
The arc of the largest semicircle is <math>8.5 \pi</math>, so if it were a full circle, the circumference would be <math>17 \pi</math>. So the <math>\text{diameter}=17</math>.
 +
 
 +
 
 +
 
 +
By the Pythagorean theorem, the other side has length <math>15</math>, so the radius is <math>\boxed{\textbf{(B)}\ 7.5}</math>
 +
 
 +
~Edited by Theraccoon to correct typos.
 +
 
 +
==Brief Explanation==
 +
SavannahSolver got a diameter of 17 because the given arc length of the semicircle was
 +
8.5π. The arc length of a semicircle can be calculated using the formula
 +
πr, where
 +
r is the radius. let’s use the full circumference formula for a circle, which is
 +
2πr. Since the semicircle is half of a circle, its arc length is
 +
πr, which was given as
 +
8.5π. Solving for
 +
r, we get
 +
𝑟=8.5
 +
. Therefore, the diameter, which is
 +
2r, is
 +
2x8.5=17
 +
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>
 +
 
 +
 
 +
. - TheNerdWhoIsNerdy.
  
 
==Solution 2==
 
==Solution 2==
−
We go as in Solution 1, finding the diameter of the circle on AC and 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 <math>\frac{289\pi}{8}</math>, and the middle one is <math>\frac{289\pi}{8}-\frac{64\pi}{8}=\frac{225\pi}{8}</math>, so the radius is <math>\frac{15}{2}=\boxed{\textbf{(B)}\ 7.5}</math>.
+
We go as in Solution 1, finding the diameter of the circle on <math>\overline{AC}</math> and <math>\overline{AB}</math>. 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 <math>\frac{289\pi}{8}</math>, and the middle one is <math>\frac{289\pi}{8}-\frac{64\pi}{8}=\frac{225\pi}{8}</math>, so the radius is <math>\frac{15}{2}=\boxed{\textbf{(B)}\ 7.5}</math>.
 +
 
 +
~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==
  
−
==See Also==
+
~ Mia Wang the Author
−
{{AMC8 box|year=2013|num-b=22|num-a=24}}
+
~skibidi
−
{{MAA Notice}}
 

Latest revision as of 22:26, 18 January 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 8.5π. The arc length of a semicircle can be calculated using the formula πr, where r is the radius. let’s use the full circumference formula for a circle, which is 2πr. Since the semicircle is half of a circle, its arc length is πr, which was given as 8.5π. Solving for r, we get 𝑟=8.5 . Therefore, the diameter, which is 2r, is 2x8.5=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.

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

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