Difference between revisions of "1998 AHSME Problems/Problem 26"

Revision as of 21:05, 9 February 2008

Problem

In quadrilateral $ABCD$ , it is given that $\angle A = 120^{\circ}$ , angles $B$ and $D$ are right angles, $AB = 13$ , and $AD = 46$ . Then $AC=$ $\mathrm{(A)}\ 60 \qquad\mathrm{(B)}\ 62 \qquad\mathrm{(C)}\ 64 \qquad\mathrm{(D)}\ 65 \qquad\mathrm{(E)}\ 72$

Solution

Solution 1

Let the extensions of $\overline{DA}$ and $\overline{CB}$ be at $E$ . Since $\angle BAD = 120^{\circ}$ , $\angle BAE = 60^{\circ}$ and $\triangle ABE$ is a 30-60-90 triangle. Also, $\triangle ABE \sim \triangle CDE$ , so $\triangle CDE$ is also a 30-60-90 triangle.

$[asy] size(200); defaultpen(0.8); pair D=(0,0), C=(0,24*3^0.5), A=(46,0), E=(72,0), B=(46+13/2,13*3^.5/2); pair P=(C+D)/2, Q=(D+A)/2, R=(A+E)/2, T=(A+B)/2; draw(D--A--B--C--cycle); draw(C--A); draw(A--E--B,dashed); label("$A$",A,SSW); label("$B$",B,NNE); label("$C$",C,WNW); label("$D$",D,SSW); label("$E$",E,SSE); label("24$\sqrt{3}$",P,W); label("46",Q,S); label("26",R,S); label("13",T,WNW); [/asy]$

Thus $AE = 2AB = 26$ , and $CD = \frac{26 + 46}{\sqrt{3}} = 24\sqrt{3}$ . By the Pythagorean Theorem on $\triangle ACD$ , $AC = \sqrt{(46)^2 + (24\sqrt{3})^2} = 62 \Rightarrow \mathrm{(B)}.$

Solution 2

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Opposite angles add up to $180^{\circ}$ , so $ABCD$ is a cyclic quadrilateral. Also, $\angle B = \angle D = 90^{\circ}$ , from which it follows that $\overline{AC}$ is a diameter of the circumscribing circle. We can apply the extended version of the Law of Sines on $\triangle ABD$ :

$AC = 2R = \frac{BD}{\sin 120^{\circ}} = \frac{2}{\sqrt{3}} BD$

By the Law of Cosines on $\triangle ABD$ :

$BD^2 = 13^2 + 46^2 - 2 \cdot 13 \cdot 46 \cdot \cos 120^{\circ} = 2883$

So $AC = \frac{2}{\sqrt{3}} \sqrt{2883} = 62$ .

@@ Line 25: / Line 25: @@
 label("\(D\)",D,SSW);
 label("\(E\)",E,SSE);
-label("24<math>\sqrt{3}</math>",P,W);
+label("24\(\sqrt{3}\)",P,W);
 label("46",Q,S);
 label("26",R,S);

1998 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 25	Followed by Problem 27
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