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

Answer: $(C) 2 \sqrt{6}$

Given, a 14-9-7-12 quadrilateral ( which has an in-circle).

Find the largest possible in-radius.

Solution:

Since Area = $r \times$ semi-perimeter, and perimeter is fixed, we can maximize the area. Let the angle between the 14 and 12 be $\alpha$ degree, and the one between the 9 and 7 be $\beta$ .

2(Area) = $(14)(12) \sin \alpha + (9)(7) \sin \beta$

$\frac{2}{21}$ (Area) = $8 \sin \alpha + 3 \sin \beta$

By law of cosine, $14^2 + 12 ^2 - 2(14)(12) \cos \alpha = 9^2 + 7^2 - 2(9)(7) \cos \beta$

$8 \cos \alpha - 3 \cos \beta = 5$ (simple algebra left to the reader)

$\frac{4}{441}$ (Area) $^2 + 25$ = $64 \sin^2 \alpha + 9 \sin^2 \beta + 64 \cos^2 \alpha + 9 \cos^2 \beta - 48 \cos \alpha \cos \beta + 48 \sin \alpha \sin \beta$

$= 73 - 48 \cos (\alpha + \beta)$

$\frac{4}{441}$ (Area) $^2$ = $48 ( 1- \cos (\alpha + \beta))$ , which reaches maximum when $( 1- \cos (\alpha + \beta)) = 2$ .

(and since it is a quadrilateral, it is possible to have $\alpha + \beta = \pi$ (hence cyclic quadrilateral, that would be the best guess and the Brahmagupta's formula would work for area and the work is simple).

$\frac{4}{441}$ (Area) $^2 \le 96$

(Area) $^2 \le 24 (441)$

(Area) $\le 42 \sqrt{6}$ , Area = $r \times$ semi-perimeter.

Hence, $r = 2 \sqrt{6}$ , choice $(C)$

2011 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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2011 AMC 12A Problems/Problem 24

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