Difference between revisions of "2017 AMC 12A Problems/Problem 16"

Revision as of 21:23, 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

$[asy] size(7.5cm); draw(arc((0,0),3,0,180)); draw(arc((2,0),1,0,180)); draw(arc((-1,0),2,0,180)); draw((-3,0)--(3,0)); pair P = (9/7,12/7); pair A = (-1,0); pair C = (0,0); pair B = (2,0); draw(circle(P,6/7)); dot((-1,0)); dot((2,0)); dot((0,0)); dot(P); draw((-1,0)--P); draw((-2.45,12/7)--P); draw((2.45,12/7)--P); draw((2,0)--P); draw((0,0)--(9/5,12/5)); label("$A$",A,SW); label("$C$",C,S); label("$B$",B,SE); label("$P$",P,N); [/asy]$

Let $C$ be the center of the largest semicircle. We know that $AC = 1$ , $CB = 2$ , $AP = r + 2$ , $BP = r + 1$ , and $CP = 3 - r$ . Notice that $\Delta ACP$ and $\Delta CBP$ are bounded by the same two parallel lines, so these triangles have the same heights. Because the bases of these two triangles (that have the same heights) differ by a factor of 2, the area of $\Delta CBP$ must be twice that of $\Delta ACP$ , since the area of a triangle is $\frac{1}{2} Base \cdot Height$ .

Again, we don't need to look at the circle and the semicircles anymore; just focus on the triangles.

$[asy] size(7.5cm); pair A = (-1,0); pair C = (0,0); pair B = (2,0); draw((-1,0)--(2,0)); pair P = (9/7,12/7); dot((-1,0)); dot((2,0)); dot((0,0)); dot(P); draw((-1,0)--P); draw((2,0)--P); draw((0,0)--P); label("$A$",A,W); label("$C$",C,S); label("$B$",B,E); label("$P$",P,N); [/asy]$

Let $A_1$ equal to the area of $\Delta ACP$ and $A_2$ equal to the area of $\Delta CBP$ . 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.

The semiperimeter $s_1$ of $\Delta ACP$ is $\frac{[(r + 2) + (3 - r) + 1]}{2} = \frac{6}{2} = 3$ 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 Heron's Formula again, find the area of $\Delta CBP$ with sides $r+1$ , $2$ , and $3-r$ .

$s_2 = \frac{(r + 1) + 2 + (3 - r)}{2} = 3$ $A_2 = \sqrt{3(3-2)(3-1-r)(3-3+r)} = \sqrt{3(2r-r^2)} = \sqrt{6r-3r^2}$

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

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