Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(→‎Solution 2)
(→‎Solution 3)
Line 115: Line 115:
 
<asy>
 
<asy>
 
size(5cm);
 
size(5cm);
 +
pair A = (-1,0);
 +
pair C = (0,0);
 +
pair B = (2,0);
 
draw((-1,0)--(2,0));
 
draw((-1,0)--(2,0));
 
pair P = (9/7,12/7);
 
pair P = (9/7,12/7);
Line 121: Line 124:
 
draw((2,0)--P);
 
draw((2,0)--P);
 
draw((0,0)--P);
 
draw((0,0)--P);
 +
label("$A$",A,W);
 +
label("$C$",C,S);
 +
label("$B$",B,E);
 +
label("$P$",P,N);
 
</asy>
 
</asy>
  

Revision as of 21:02, 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); 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]

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_12A_Problems/Problem_16&oldid=83196"