Difference between revisions of "1992 AIME Problems/Problem 9"

Revision as of 20:56, 6 November 2017

Problem

Trapezoid $ABCD^{}_{}$ has sides $AB=92^{}_{}$ , $BC=50^{}_{}$ , $CD=19^{}_{}$ , and $AD=70^{}_{}$ , with $AB^{}_{}$ parallel to $CD^{}_{}$ . A circle with center $P^{}_{}$ on $AB^{}_{}$ is drawn tangent to $BC^{}_{}$ and $AD^{}_{}$ . Given that $AP^{}_{}=\frac mn$ , where $m^{}_{}$ and $n^{}_{}$ are relatively prime positive integers, find $m+n^{}_{}$ .

Solution 1

Let $AB$ be the base of the trapezoid and consider angles $A$ and $B$ . Let $x=AP$ and let $h$ equal the height of the trapezoid. Let $r$ equal the radius of the circle.

Then

$\sin{A}= \frac{r}{x} = \frac{h}{70}\qquad\text{ and }\qquad\sin{B}= \frac{r}{92-x} = \frac{h}{50}.\tag{1}$

Let $z$ be the distance along $AB$ from $A$ to where the perp from $D$ meets $AB$ .

Then $h^2 +z^2 =70^2$ and $(73-z)^2 + h^2 =50^2$ so $h =\frac{\sqrt{44710959}}{146}$ . We can substitute this into $(1)$ to find that $x= \frac{11753}{219} = \frac{161}{3}$ and $m+n = 164$ .

Remark: One can come up with the equations in $(1)$ without directly resorting to trig. From similar triangles, $h/r = 70/x$ and $h/r = 50/ (92-x)$ . This implies that $70/x =50/(92-x)$ , so $x = 161/3$ .

Solution 2

From $(1)$ above, $x = \frac{70r}{h}$ and $92-x = \frac{50r}{h}$ . Adding these equations yields $92 = \frac{120r}{h}$ . Thus, $x = \frac{70r}{h} = \frac{7}{12}\cdot\frac{120r}{h} = \frac{7}{12}\cdot92 = \frac{161}{3}$ , and $m+n = \boxed{164}$ .

We can use $(1)$ from Solution 1 to find that $h/r = 70/x$ and $h/r = 50/ (92-x)$ .

This implies that $70/x =50/(92-x)$ so $x = 161/3$

Solution 3

Extend $AD$ and $BC$ to meet at a point $X$ . Since $AB$ and $CD$ are parallel, $\triangle XCD ~ \triangle XAB$ . If $AX$ is further extended to a point $A'$ and $XB$ is extended to a point $B'$ such that $A'B'$ is tangent to circle $P$ , we discover that circle $P$ is the incircle of triangle $XA'B'$ . Then line $XP$ is the angle bisector of $\angle AXB$ . By homothety, $P$ is the intersection of the angle bisector of $\triangle XAB$ with $AB$ . By the angle bisector theorem,

$\begin{align*} \frac{AX}{AP} &= \frac{XB}{BP}\\ \frac{AX}{AP} - \frac{XD}{AP} &= \frac{XB}{BP} - \frac{XC}{BP}\\ \frac{AD}{AP} &= \frac{BC}{BP}\\ &=\frac{7}{5} \end{align*}$

Let $7a = AP$ , then $AB = 7a + 5a = 12a$ . $AP = \frac{7}{12}(AB) = \frac{92\times 7}{12} = \frac{161}{3}$ . Thus, $m + n = 164$ .

Note: this solution shows that the length of $CD$ is irrelevant as long as there still exists a circle as described in the problem.

Solution 4

The area of the trapezoid is $\frac{(19+92)h}{2}$ , where $h$ is the height of the trapezoid.

Draw lines $CP$ and $BP$ . We can now find the area of the trapezoid as the sum of the areas of the three triangles $BPC$ , $CPD$ , and $PBA$ .

$[BPC] = \frac{1}{2} \cdot 50 \cdot r$ (where $r$ is the radius of the tangent circle.)

$[CPD] = \frac{1}{2} \cdot 19 \cdot h$

$[PBA] = \frac{1}{2} \cdot 70 \cdot r$

$[BPC] + [CPD] + [PBA] = 60r + \frac{19h}{2} = [ABCD] = \frac{(19+92)h}{2}$

$60r = 46h$

$r = \frac{23h}{30}$

From Solution 1 above, $\frac{h}{70} = \frac{r}{x}$

Substituting $r = \frac{23h}{30}$ , we find $x = \frac{161}{3}$ , hence the answer is $\boxed{164}$ .

@@ Line 35: / Line 35: @@
 Let <math>7a = AP</math>, then <math>AB = 7a + 5a = 12a</math>. <math>AP = \frac{7}{12}(AB) = \frac{92\times 7}{12} = \frac{161}{3}</math>. Thus, <math>m + n = 164</math>.
+Note: this solution shows that the length of <math>CD</math> is irrelevant as long as there still exists a circle as described in the problem.
 ==Solution 4==

1992 AIME (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
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