2011 AMC 12A Problems/Problem 24

Problem

Consider all quadrilaterals $ABCD$ such that $AB=14$ , $BC=9$ , $CD=7$ , and $DA=12$ . What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?

$\textbf{(A)}\ \sqrt{15} \qquad \textbf{(B)}\ \sqrt{21} \qquad \textbf{(C)}\ 2\sqrt{6} \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 2\sqrt{7}$

Solution

Solution 1

Note as above that ABCD must be cyclic to obtain the circle with maximal radius. Let $E$ , $F$ , $G$ , and $H$ be the points on $AB$ , $BC$ , $CD$ , and $DA$ respectively where the circle is tangent. Let $\theta=\angle BAD$ and $\alpha=\angle ADC$ . Since the quadrilateral is cyclic, $\angle ABC=180^{\circ}-\alpha$ and $\angle BCD=180^{\circ}-\theta$ . Let the circle have center $O$ and radius $r$ . Note that $OHD$ , $OGC$ , $OFB$ , and $OEA$ are right angles.

Hence $FOG=\theta$ , $GOH=180^{\circ}-\alpha$ , $EOH=180^{\circ}-\theta$ , and $FBE=\alpha$ .

Therefore, $AEOH\sim OFCG$ and $EBFO\sim HOGD$ .

Let $x=CG$ . Then $CF=x$ , $BF=BE=9-x$ , $GD=DH=7-x$ , and $AH=AE=x+5$ . Using $AEOH\sim OFCG$ and $EBFO\sim HOGD$ we have $r/(x+5)=x/r$ , and $(9-x)/r=r/(7-x)$ . By equating the value of $r^2$ from each, $x(x+5)=(7-x)(9-x)$ . Solving we obtain $x=3$ so that $\boxed{\textbf{(C)}\ 2\sqrt{6}}$ .

Solution 2

To maximize the radius of the circle, we also need to maximize the area. To maximize the area of the circle, the quadrilateral must be tangential (have an incircle). In a tangential quadrilateral, the sum of opposite sides is equal to the semiperimeter of the quadrilateral. So, $14+7=12+9$ . Therefore, it has an incircle. By definition, given $4$ side lengths, a cyclic quadrilateral has the maximum area of any quadrilateral with those side lengths. Therefore, to maximize the area of the quadrilateral and thus the incircle, we assume that this quadrilateral is cyclic.

For cyclic quadrilaterals, the area is $\sqrt{(s-a)(s-b)(s-c)(s-d)}$ where $s$ is the semiperimeter of the cyclic quadrilateral and $a, b, c,$ and $d$ are the sides of the quadrilateral. Compute this area to get $42\sqrt{6}$ . The area of a tangential quadrilateral is given by the $rs$ formula, where $rs=A$ . We can substitute $s$ , the semiperimeter, and $A$ , the area and solve for r to get $\boxed{\textbf{(C)}\ 2\sqrt{6}}$ .

2011 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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

Page

Toolbox

Search

2011 AMC 12A Problems/Problem 24

Contents

Problem

Solution

Solution 1

Solution 2

See also