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Difference between revisions of "2011 AMC 12A Problems/Problem 24"

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== Solution ==
 
== Solution ==
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=== Solution 1 ===
  
 
Since Area = <math>r \times</math> semi-perimeter, and perimeter is fixed, we can maximize the area. Let the angle between the 14 and 12 be <math>\alpha</math> degree, and the one between the 9 and 7 be <math>\beta</math>.
 
Since Area = <math>r \times</math> semi-perimeter, and perimeter is fixed, we can maximize the area. Let the angle between the 14 and 12 be <math>\alpha</math> degree, and the one between the 9 and 7 be <math>\beta</math>.
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Hence, <math>r = 2 \sqrt{6}</math>, choice <math>(C)</math>
 
Hence, <math>r = 2 \sqrt{6}</math>, choice <math>(C)</math>
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=== Solution 2 ===
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To maximize the radius of the circle, we also maximize the area. To maximize the area of the circle, the quadrilateral must be tangential (have an incircle). A tangential quadrilateral has the property that the sum of a of opposite sides is equal to the semiperimeter of the quadrilateral. In this case, <math>14+7=12+9</math>. Therefore, it has an incircle. By definition, a cyclic quadrilateral has the maximum area for a quadrilateral with corresponding side lengths. Therefore, to maximize the area of the tangential quadrilateral and thus the incircle, we assume that this quadrilateral is cyclic. For cyclic quadrilaterals, the area is given by <math>\sqrt{(s-a)(s-b)(s-c)(s-d)}</math> where <math>s</math> is the semiperimeter of the cyclic quadrilateral and <math>a, b, c,</math> and <math>d</math> are the sides of the quadrilateral. Compute this area to get <math>42\sqrt{6}</math>. The area of a tangential quadrilateral is given by the <math>rs</math> formula, where <math>rs=A</math>. We can substitute <math>s</math>, the semiperimeter, and <math>A</math>, and area, for their respective values and solve for r to get <math>\boxed{\textbf{(C)}\ 2\sqrt{6}</math>.
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2011|num-b=23|num-a=25|ab=A}}
 
{{AMC12 box|year=2011|num-b=23|num-a=25|ab=A}}

Revision as of 02:32, 20 February 2011

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

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 $(C)$

Solution 2

To maximize the radius of the circle, we also maximize the area. To maximize the area of the circle, the quadrilateral must be tangential (have an incircle). A tangential quadrilateral has the property that the sum of a of opposite sides is equal to the semiperimeter of the quadrilateral. In this case, $14+7=12+9$. Therefore, it has an incircle. By definition, a cyclic quadrilateral has the maximum area for a quadrilateral with corresponding side lengths. Therefore, to maximize the area of the tangential quadrilateral and thus the incircle, we assume that this quadrilateral is cyclic. For cyclic quadrilaterals, the area is given by $\sqrt{(s-a)(s-b)(s-c)(s-d)}$ 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 $A$, and area, for their respective values and solve for r to get $\boxed{\textbf{(C)}\ 2\sqrt{6}$ (Error compiling LaTeX. Unknown error_msg).

See also

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
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