2016 AMC 12A Problems/Problem 15

Problem

Circles with centers $P, Q$ and $R$ , having radii $1, 2$ and $3$ , respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$ , respectively, with $Q'$ between $P'$ and $R'$ . The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$ ?

$\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{6}/3\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{6}/2$

Solution 1

$[asy] size(250); defaultpen(linewidth(0.4)); //Variable Declarations pair P,Q,R,Pp,Qp,Rp; pair A,B; //Variable Definitions A=(-5, 0); B=(8, 0); P=(-2.828,1); Q=(0,2); R=(4.899,3); Pp=foot(P,A,B); Qp=foot(Q,A,B); Rp=foot(R,A,B); path PQR = P--Q--R--cycle; //Initial Diagram dot(P); dot(Q); dot(R); dot(Pp); dot(Qp); dot(Rp); draw(Circle(P, 1), linewidth(0.8)); draw(Circle(Q, 2), linewidth(0.8)); draw(Circle(R, 3), linewidth(0.8)); draw(A--B,Arrows); label("$P$",P,N); label("$Q$",Q,N); label("$R$",R,N); label("$P'$",Pp,S); label("$Q'$",Qp,S); label("$R'$",Rp,S); label("$l$",B,E); //Added lines draw(PQR); draw(P--Pp); draw(Q--Qp); draw(R--Rp); //Angle marks draw(rightanglemark(P,Pp,B)); draw(rightanglemark(Q,Qp,B)); draw(rightanglemark(R,Rp,B)); [/asy]$

Notice that we can find $[P'PQRR']$ in two different ways: $[P'PQQ']+[Q'QRR']$ and $[PQR]+[P'PRR']$ , so $[P'PQQ']+[Q'QRR']=[PQR]+[P'PRR']$ $\break$

$P'Q'=\sqrt{PQ^2-(QQ'-PP')^2}=\sqrt{3^2-1^2}=\sqrt{8}=2\sqrt{2}$ . Additionally, $Q'R'=\sqrt{QR^2-(RR'-QQ')^2}=\sqrt{5^2-1^2}=\sqrt{24}=2\sqrt{6}$ . Therefore, $[P'PQQ']=\frac{P'P+Q'Q}{2}*2\sqrt{2}=\frac{1+2}{2}*2\sqrt{2}=3\sqrt{2}$ . Similarly, $[Q'QRR']=5\sqrt6$ . We can calculate $[P'PRR']$ easily because $P'R'=P'Q'+Q'R'=2\sqrt{2}+2\sqrt{6}$ . $[P'PRR']=4\sqrt{2}+4\sqrt{6}$ . $\newline$

Plugging into first equation, the two sums of areas, $3\sqrt{2}+5\sqrt{6}=4\sqrt{2}+4\sqrt{6}+[PQR]$ . $\newline$

$[PQR]=$ \boxed{\textbf{(D) }\sqrt{6}-\sqrt{2}.}$==Solution 2==

Use the [[Shoelace Theorem]].

Let the center of the first circle of radius 1 be at$ (Error compiling LaTeX. Unknown error_msg)(0, 1)$.

Draw the trapezoid$ (Error compiling LaTeX. Unknown error_msg)PQQ'P' $and using the Pythagorean Theorem, we get that$ P'Q' = 2\sqrt{2} $so the center of the second circle of radius 2 is at$ (2\sqrt{2}, 2)$.

Draw the trapezoid$ (Error compiling LaTeX. Unknown error_msg)QRR'Q' $and using the Pythagorean Theorem, we get that$ Q'R' = 2\sqrt{6} $so the center of the third circle of radius 3 is at$ (2\sqrt{2}+2\sqrt{6}, 3)$.

Now, we may use the Shoelace Theorem!$ (Error compiling LaTeX. Unknown error_msg)(0,1)$$ (Error compiling LaTeX. Unknown error_msg)(2\sqrt{2}, 2)$$ (Error compiling LaTeX. Unknown error_msg)(2\sqrt{2}+2\sqrt{6}, 3)$$ (Error compiling LaTeX. Unknown error_msg)\frac{1}{2}|(2\sqrt{2}+4\sqrt{2}+4\sqrt{6})-(6\sqrt{2}+2\sqrt{2}+2\sqrt{6})|$$ (Error compiling LaTeX. Unknown error_msg)= \sqrt{6}-\sqrt{2}$$ (Error compiling LaTeX. Unknown error_msg)\fbox{D}$.

==Solution 3==$ (Error compiling LaTeX. Unknown error_msg)PQ = 3 $and$ QR = 5 $because they are the sum of two radii.$ QQ' - PP' = 1 $and$ RR' - QQ' = 1 $, the difference of the radii. Using pythagorean theorem, we find that$ P'Q' $and$ Q'R' $are$ \sqrt{8} $and$ \sqrt{24} $,$ P'R' = \sqrt{8} + \sqrt{24}$.

Draw a perpendicular from$ (Error compiling LaTeX. Unknown error_msg)P $to line$ RR' $, then we can use the Pythagorean theorem to find$ PR $.$ RR' - PP' = 2$. We get <cmath>PR^2 = (\sqrt{8} + \sqrt{24})^2 + 4 = 36 + 16\sqrt{3} \Rightarrow PR = \sqrt{36 + 16\sqrt{3}} = 2\sqrt{9 + 4\sqrt{3}}</cmath>

To make our calculations easier, let$ (Error compiling LaTeX. Unknown error_msg)\sqrt{9 + 4\sqrt{3}} = a $. The semi-perimeter of our triangle is$ \frac{3 + 5 + 2a}{2} = 4 + a $. Symbolize the area of the triangle with$ A$. Using Heron's formula, we have <cmath>A^2 = (4 + a)(4 + a - 2a)(4 + a - 3)(4 + a - 5) = (4 + a)(4 - a)(a + 1)(a - 1) = (16 - a^2)(a^2 - 1)</cmath> We can remove the outer root of a. <cmath>A^2 = (16 - 9 - 4\sqrt{3})(9 + 4\sqrt{3} - 1) = (7 - 4\sqrt{3})(8 + 4\sqrt{3}) = 8 - 4\sqrt{3} \rightarrow A = \sqrt{8 - 4\sqrt{3}}</cmath>

We solve the nested root. We want to turn$ (Error compiling LaTeX. Unknown error_msg)8 - 4\sqrt{3} $into the square of something. If we have$ (a - b) ^ 2 = 8 - 4\sqrt{3} $, then we get <cmath>\begin{cases} a^2 + b^2 = 8 \\ ab = 2\sqrt{3} \end{cases}</cmath> Solving the system of equations, we get$ a = \sqrt{6} $and$ b = \sqrt{2} $. Alternatively, you can square all the possible solutions until you find one that is equal to$ 8 - 4\sqrt{3}$. $A = \sqrt{8 - 4\sqrt{3}} = \sqrt{(\sqrt{6} - \sqrt{2})^2} = \sqrt{6} - \sqrt{2} \rightarrow \fbox{D}$ ~ZericH

Solution 4

$[asy] // Initial Pen Sizing size(250); defaultpen(linewidth(0.4)); defaultpen(fontsize(10pt)); // Variable Declarations pair P,Q,R,Pp,Qp,Rp,X,Y,Z,A,B; // Variable Definitions A=(-5, 0); B=(8, 0); P=(-2.828,1); Q=(0,2); R=(4.899,3); X=(0,1); Y=(4.899,1); Z=(4.899,2); Pp=foot(P,A,B); Qp=foot(Q,A,B); Rp=foot(R,A,B); path PQR = P--Q--R--cycle; //Initial Diagram dot(P); dot(Q); dot(R); dot(Pp); dot(Qp); dot(Rp); dot(X); dot(Y); dot(Z); draw(Circle(P, 1), linewidth(0.3)); draw(Circle(Q, 2), linewidth(0.3)); draw(Circle(R, 3), linewidth(0.3)); draw(A--B,Arrows); label("$P$",P,N); label("$Q$",Q,N); label("$R$",R,N); label("$P'$",Pp,S); label("$Q'$",Qp,S); label("$R'$",Rp,S); label("$l$",B,E); label("$X$",X,NE); label("$Y$",Y,E); label("$Z$",Z,E); //Added lines filldraw(PQR,gray(0.8)); draw(P--Pp,linetype("8 8")); draw(Q--Qp,linetype("8 8")); draw(R--Rp,linetype("8 8")); draw(P--Y,linetype("8 8")); draw(Q--Z,linetype("8 8")); //Angle marks draw(rightanglemark(R,Y,P)); draw(rightanglemark(Q,X,P)); draw(rightanglemark(R,Z,Q)); //Length labeling label("$2\sqrt{2}$",P--X,fontsize(8pt)); label("$2\sqrt{6}$",X--Y,fontsize(8pt)); label("$2\sqrt{6}$",Q--Z,fontsize(8pt)); label("$1$",R--Z,E,fontsize(8pt)); label("$1$",Z--Y,E,fontsize(8pt)); label("$3$",(-2.828,1.3)--Q,W,fontsize(8pt)); label("$5$",Q--R,N,fontsize(8pt)); [/asy]$ The above diagram can be achieved relatively simply using basic knowledge of the Pythagorean theorem and the fact that the radius from the center to the point of tangency is perpendicular to the tangent line. From there, observe that $[PQRY]$ can be calculated in two ways: $[\triangle PQX] + [QZYX] + [\triangle QRZ]$ and $[\triangle PRY] + [\triangle PQR]$ . Solving, we get: $\begin{align*} [\triangle PQR] &= [PQRY] - [\triangle PRY] \\ &= [\triangle PQX] + [QZYX] + [\triangle QRZ] - [\triangle PRY] \\ &= \sqrt{2} + 2\sqrt{6} + \sqrt{6}- 2\sqrt{2}-2\sqrt{6} \\ &= \boxed{\textbf{(D)} \sqrt{6}-\sqrt{2}} \end{align*}$

- ColtsFan10, diagram partially borrowed from Solution 1

Solution 5 (Heron’s)

We can use the Pythagorean theorem to find that the lengths are $3,5,\sqrt{36+16\sqrt{3}}$ . If we apply Heron’s, we know that it must be the sum (or difference) of two or more square roots, by instinct. This means that $\boxed{\textbf{(D)} \sqrt{6}-\sqrt{2}}$ is the answer.

Solution 6 (Educated Guess)

Like Solution 1, we can use the Pythagorean theorem to find $P'Q'$ and $Q'R'$ , which are $2\sqrt2$ and $2\sqrt6$ respectively. Since the only answer choice that has $\sqrt2$ and $\sqrt6$ is $D$ , we can make an educated guess that $\boxed{\textbf{(D)} \sqrt{6}-\sqrt{2}}$ is the answer.

Video Solutions

https://www.youtube.com/watch?v=sWurLJqf02Y ~by Punxsutawney Phil

https://www.youtube.com/watch?v=UanfIBpDTh8&ab_channel=ArtofProblemSolving ~Art of Problem Solving

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