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Difference between revisions of "2017 AMC 12A Problems/Problem 16"
(→‎Solution 2)
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<math>8r + 2 = -20r + 26</math>
 
<math>8r + 2 = -20r + 26</math>
 
<math>28r = 24</math>, so <math>r</math> = <math>6/7</math> <math>\boxed{(B)}</math>
 
<math>28r = 24</math>, so <math>r</math> = <math>6/7</math> <math>\boxed{(B)}</math>
 +
 +
==Solution 3==
 +
 +
<asy>
 +
size(5cm);
 +
draw(arc((0,0),3,0,180));
 +
draw(arc((2,0),1,0,180));
 +
draw(arc((-1,0),2,0,180));
 +
draw((-3,0)--(3,0));
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pair P = (9/7,12/7);
 +
draw(circle(P,6/7));
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dot((-1,0)); dot((2,0)); dot((0,0)); dot(P);
 +
draw((-1,0)--P);
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draw((2,0)--P);
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draw((0,0)--(9/5,12/5));
 +
</asy>
 +
 +
Let <math>C</math> be the center of the largest semicircle. Notice that <math>\Delta ACP</math> and <math>\Delta CBP</math> are bounded by the same two parallel lines and therefore have the same heights. Since the bases of these two triangles differ by a factor of 2, the area of <math>\Delta CBP</math> must be twice that of <math>\Delta ACP</math>.
 +
We know that <math>AC</math> <math>=</math> 1,  <math>CB</math> <math>=</math> 2,  <math>AP</math> <math>=</math> <math>r</math> <math>+</math> 2,  <math>BP</math> <math>=</math> <math>r</math> <math>+</math> 1,  <math>CP</math> <math>=</math> <math>3</math> <math>-</math> <math>r</math>.
 +
Again, we don't even need the circles and semicircles anymore; just focus on the triangles.
 +
 +
<asy>
 +
size(5cm);
 +
draw((-1,0)--(2,0));
 +
pair P = (9/7,12/7);
 +
dot((-1,0)); dot((2,0)); dot((0,0)); dot(P);
 +
draw((-1,0)--P);
 +
draw((2,0)--P);
 +
draw((0,0)--P);
 +
</asy>
 +
 +
The semiperimeter <math>s_1</math> of <math>\Delta ACP</math> is <cmath>[(r + 2) + (3 - r) + 1] / 2 = 6/2 = 3</cmath>
 +
 +
[[Heron's Formula]] states that the area of an triangle with sides <math>a</math> <math>b</math> and <math>c</math> is
 +
<cmath>\sqrt{s(s-a)(s-b)(s-c)}</cmath>
 +
where <math>s</math>, or the semiperimeter, is half of the triangle's perimeter.
 +
 +
Use Heron's Formula to obtain
 +
 +
<cmath>A_1 = \sqrt{3(2)(3-2-r)(3-3+r)} = \sqrt{6r(1-r)} = \sqrt{6r-6r^2}</cmath>
 +
 +
Using the same formula again, find the area of the second, larger triangle (<math>\Delta CBP</math>) with sides <math>r+1</math>, <math>2</math>, and <math>3-r</math>.
 +
 +
<cmath>s_2 = ((r + 1) + 2 + (3 - r)) / 2 = 3</cmath>
 +
<cmath>A_2 = \sqrt{3(3-2)(3-1-r)(3-3+r)} = \sqrt{6(2r-r^2)} = \sqrt{6r-3r^2}</cmath>
 +
 +
 +
Now,
 +
<cmath>A_2 = 2 \cdot A_1</cmath>
 +
<cmath>2\sqrt{6r-6r^2} = \sqrt{6r-3r^2}</cmath>
 +
<cmath>4(6r-6r^2) = 6r-3r^2</cmath>
 +
<cmath>24r-24r^2 = 6r-3r^2</cmath>
 +
<cmath>18r = 21r^2</cmath>
 +
<cmath>r = \frac{18}{21} = \boxed{\frac{6}{7}} \to \boxed{\textbf{(B)}}</cmath>
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2017|ab=A|num-b=15|num-a=17}}
 
{{AMC12 box|year=2017|ab=A|num-b=15|num-a=17}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 20:48, 8 February 2017

Problem

In the figure below, semicircles with centers at $A$ and $B$ and with radii 2 and 1, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter $JK$. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at $P$ is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at $P$?

[asy] size(5cm); draw(arc((0,0),3,0,180)); draw(arc((2,0),1,0,180)); draw(arc((-1,0),2,0,180)); draw((-3,0)--(3,0)); pair P = (-1,0)+(2+6/7)*dir(36.86989); draw(circle(P,6/7)); dot((-1,0)); dot((2,0)); dot(P); [/asy]

$\textbf{(A)}\ \frac{3}{4} \qquad \textbf{(B)}\ \frac{6}{7} \qquad\textbf{(C)}\ \frac{1}{2}\sqrt{3} \qquad\textbf{(D)}\ \frac{5}{8}\sqrt{2} \qquad\textbf{(E)}\ \frac{11}{12}$

Solution 1

Connect the centers of the tangent circles! (call the center of the large circle $C$)

[asy] size(5cm); draw(arc((0,0),3,0,180)); draw(arc((2,0),1,0,180)); draw(arc((-1,0),2,0,180)); draw((-3,0)--(3,0)); pair P = (9/7,12/7); draw(circle(P,6/7)); dot((-1,0)); dot((2,0)); dot((0,0)); dot(P); draw((-1,0)--P); draw((2,0)--P); draw((0,0)--(9/5,12/5)); [/asy]

Notice that we don't even need the circles anymore; thus, draw triangle $\Delta ABP$ with cevian $PC$:

[asy] size(5cm); draw((-1,0)--(2,0)); pair P = (9/7,12/7); dot((-1,0)); dot((2,0)); dot((0,0)); dot(P); draw((-1,0)--P); draw((2,0)--P); draw((0,0)--P); [/asy]

and use Stewart's Theorem:

\[AB \cdot AC \cdot BC + AB \cdot {CP}^2 = AC \cdot {BP}^2 + BC \cdot {AP}^2\]

From what we learned from the tangent circles, we have $AB = 3$, $AC = 1$, $BC = 2$, $AP = 2 + r$, $BP = 1 + r$, and $CP = 3 - r$, where $r$ is the radius of the circle centered at $P$ that we seek.

Thus:

\[3 \cdot 1 \cdot 2 + 3 {\left(3-r\right)}^2 = 1 {\left(1+r\right)}^2 + 2 {\left(2+r\right)}^2\] \[6 + 3\left(9 - 6r + r^2\right) = \left(1 + 2r + r^2\right) + 2\left(4 + 4r + r^2\right)\] \[33 - 18r + 3r^2 = 9 + 10r + 3r^2\] \[28r = 24\] \[r = \boxed{\frac{6}{7}} \to \boxed{\textbf{(B)}}\]

NOTICE to proficient editors: please label the points on the diagrams, thanks!

Solution 2

[asy] size(5cm); draw(arc((0,0),3,0,180)); draw(arc((2,0),1,0,180)); draw(arc((-1,0),2,0,180)); draw((-3,0)--(3,0)); pair P = (9/7,12/7); draw(circle(P,6/7)); dot((-1,0)); dot((2,0)); dot((0,0)); dot(P); draw((-1,0)--P); draw((2,0)--P); draw((0,0)--(9/5,12/5)); [/asy]

Like the solution above, connecting the centers of the circles results in triangle $\Delta ABP$ with cevian $PC$. The two triangles $\Delta APC$ and $\Delta ABP$ share angle $A$, which means we can use Law of Cosines to set up a system of 2 equations that solve for $r$ respectively:

$(2 + r)^2 + 1^2 - 2(2 + r)(1)cosA = (3 - r)^2$ (notice that the diameter of the largest semicircle is 6, so its radius is 3 and $PC$ is 3 - r)

$(2 + r)^2 + 3^2 - 2(2 + r)(3)cosA = (r+1)^2$

We can eliminate the extra variable of angle $A$ by multiplying the first equation by 3 and subtracting the second from it. Then, expand to find $r$:

$2(r^2 + 4r + 4) - 6 = 2r^2 - 20r + 26$ $8r + 2 = -20r + 26$ $28r = 24$, so $r$ = $6/7$ $\boxed{(B)}$

Solution 3

[asy] size(5cm); draw(arc((0,0),3,0,180)); draw(arc((2,0),1,0,180)); draw(arc((-1,0),2,0,180)); draw((-3,0)--(3,0)); pair P = (9/7,12/7); draw(circle(P,6/7)); dot((-1,0)); dot((2,0)); dot((0,0)); dot(P); draw((-1,0)--P); draw((2,0)--P); draw((0,0)--(9/5,12/5)); [/asy]

Let $C$ be the center of the largest semicircle. Notice that $\Delta ACP$ and $\Delta CBP$ are bounded by the same two parallel lines and therefore have the same heights. Since the bases of these two triangles differ by a factor of 2, the area of $\Delta CBP$ must be twice that of $\Delta ACP$. We know that $AC$ $=$ 1, $CB$ $=$ 2, $AP$ $=$ $r$ $+$ 2, $BP$ $=$ $r$ $+$ 1, $CP$ $=$ $3$ $-$ $r$. Again, we don't even need the circles and semicircles anymore; just focus on the triangles.

[asy] size(5cm); draw((-1,0)--(2,0)); pair P = (9/7,12/7); dot((-1,0)); dot((2,0)); dot((0,0)); dot(P); draw((-1,0)--P); draw((2,0)--P); draw((0,0)--P); [/asy]

The semiperimeter $s_1$ of $\Delta ACP$ is \[[(r + 2) + (3 - r) + 1] / 2 = 6/2 = 3\]

Heron's Formula states that the area of an triangle with sides $a$ $b$ and $c$ is \[\sqrt{s(s-a)(s-b)(s-c)}\] where $s$, or the semiperimeter, is half of the triangle's perimeter.

Use Heron's Formula to obtain

\[A_1 = \sqrt{3(2)(3-2-r)(3-3+r)} = \sqrt{6r(1-r)} = \sqrt{6r-6r^2}\]

Using the same formula again, find the area of the second, larger triangle ($\Delta CBP$) with sides $r+1$, $2$, and $3-r$.

\[s_2 = ((r + 1) + 2 + (3 - r)) / 2 = 3\] \[A_2 = \sqrt{3(3-2)(3-1-r)(3-3+r)} = \sqrt{6(2r-r^2)} = \sqrt{6r-3r^2}\]


Now, \[A_2 = 2 \cdot A_1\] \[2\sqrt{6r-6r^2} = \sqrt{6r-3r^2}\] \[4(6r-6r^2) = 6r-3r^2\] \[24r-24r^2 = 6r-3r^2\] \[18r = 21r^2\] \[r = \frac{18}{21} = \boxed{\frac{6}{7}} \to \boxed{\textbf{(B)}}\]

See Also

2017 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
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 AMC 12 Problems and Solutions

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