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

Revision as of 21:12, 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

$[asy] import olympiad; size(180); defaultpen(0.8); pair D=(0,0), C=(0,24*3^0.5), A=(46,0), B=(46+13/2,13*3^.5/2); pair P=(C+D)/2, Q=(D+A)/2, T=(A+B)/2; draw(D--A--B--C--cycle); draw(B--D,dashed); draw(A--C,dashed); draw(circumcircle(A,B,C)); label("$A$",A,SSW); label("$B$",B,NNE); label("$C$",C,WNW); label("$D$",D,SSW); label("$O$",circumcenter(A,B,C),SW); dot(circumcenter(A,B,C)); label("46",Q,S); label("13",T,E); [/asy]$

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

@@ Line 35: / Line 35: @@
 === Solution 2 ===
-{{image}}
+<center><asy>
+import olympiad;
+size(180);
+defaultpen(0.8);
+pair D=(0,0), C=(0,24*3^0.5), A=(46,0), B=(46+13/2,13*3^.5/2);
+pair P=(C+D)/2, Q=(D+A)/2, T=(A+B)/2;
+draw(D--A--B--C--cycle);
+draw(B--D,dashed);
+draw(A--C,dashed);
+draw(circumcircle(A,B,C));
+label("\(A\)",A,SSW);
+label("\(B\)",B,NNE);
+label("\(C\)",C,WNW);
+label("\(D\)",D,SSW);
+label("\(O\)",circumcenter(A,B,C),SW);
+dot(circumcenter(A,B,C));
+label("46",Q,S);
+label("13",T,E);
+</asy></center>
 Opposite angles add up to <math>180^{\circ}</math>, so <math>ABCD</math> is a cyclic quadrilateral. Also, <math>\angle B = \angle D = 90^{\circ}</math>, from which it follows that <math>\overline{AC}</math> is a diameter of the circumscribing circle. We can apply the extended version of the [[Law of Sines]] on <math>\triangle ABD</math>:
@@ Line 44: / Line 63: @@
 <cmath>BD^2 = 13^2 + 46^2 - 2 \cdot 13 \cdot 46 \cdot \cos 120^{\circ} = 2883</cmath>
-So <math>AC = \frac{2}{\sqrt{3}} \sqrt{2883} = 62</math>.
+So <math>AC = \frac{2}{\sqrt{3}} \cdot \sqrt{2883} = 62</math>.
 == See also ==

1998 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 25	Followed by Problem 27
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 • 26 • 27 • 28 • 29 • 30
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