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

Latest revision as of 09:27, 22 August 2021

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 $AP=x$ so that $PB=92-x.$ Extend $AD, BC$ to meet at $X,$ and note that $XP$ bisects $\angle AXB;$ let it meet $CD$ at $E.$ Using the angle bisector theorem, we let $XB=y(92-x), XA=xy$ for some $y.$

Then $XD=xy-70, XC=y(92-x)-50,$ thus $\frac{xy-70}{y(92-x)-50} = \frac{XD}{XC} = \frac{ED}{EC}=\frac{AP}{PB} = \frac{x}{92-x},$ which we can rearrange, expand and cancel to get $120x=70\cdot 92,$ hence $AP=x=\frac{161}{3}$ . This gives us a final answer of $161+3=\boxed{164}$

Solution 2

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 3

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 4

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 5

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 $\boxed{164}$ .

Solution 6

As the problem tells, the circle is tangent to both sides $AD,BC$ , we can make it up to a triangle $QAB$ and $P$ must lie on its angular bisector. Then we know that $AP:BP=7:5$ , which makes $AP = \frac{7}{12}(AB) = \frac{92\times 7}{12} = \frac{161}{3}$ . Thus, $m + n = 164$ .

@@ Line 1: / Line 1: @@
 == Problem ==
+Trapezoid <math>ABCD^{}_{}</math> has sides <math>AB=92^{}_{}</math>, <math>BC=50^{}_{}</math>, <math>CD=19^{}_{}</math>, and <math>AD=70^{}_{}</math>, with <math>AB^{}_{}</math> parallel to <math>CD^{}_{}</math>. A circle with center <math>P^{}_{}</math> on <math>AB^{}_{}</math> is drawn tangent to <math>BC^{}_{}</math> and <math>AD^{}_{}</math>. Given that <math>AP^{}_{}=\frac mn</math>, where <math>m^{}_{}</math> and <math>n^{}_{}</math> are relatively prime positive integers, find <math>m+n^{}_{}</math>.
-== Solution ==
+==Solution 1==
+Let <math>AP=x</math> so that <math>PB=92-x.</math> Extend <math>AD, BC</math> to meet at <math>X,</math> and note that <math>XP</math> bisects <math>\angle AXB;</math> let it meet <math>CD</math> at <math>E.</math> Using the angle bisector theorem, we let <math>XB=y(92-x), XA=xy</math> for some <math>y.</math>
+Then <math>XD=xy-70, XC=y(92-x)-50,</math> thus <cmath>\frac{xy-70}{y(92-x)-50} = \frac{XD}{XC} = \frac{ED}{EC}=\frac{AP}{PB} = \frac{x}{92-x},</cmath> which we can rearrange, expand and cancel to get <math>120x=70\cdot 92,</math> hence <math>AP=x=\frac{161}{3}</math>. This gives us a final answer of <math>161+3=\boxed{164}</math>
+== Solution 2==
+Let <math>AB</math> be the base of the trapezoid and consider angles <math>A</math> and <math>B</math>.  Let <math>x=AP</math> and let <math>h</math> equal the height of the trapezoid.  Let <math>r</math> equal the radius of the circle.
+Then
+<cmath>\sin{A}= \frac{r}{x} = \frac{h}{70}\qquad\text{ and }\qquad\sin{B}= \frac{r}{92-x}  =  \frac{h}{50}.\tag{1}</cmath>
+Let <math>z</math> be the distance along <math>AB</math> from <math>A</math> to where the perp from <math>D</math> meets <math>AB</math>.
+Then <math>h^2 +z^2 =70^2</math>  and <math>(73-z)^2 + h^2 =50^2</math>  so <math>h =\frac{\sqrt{44710959}}{146}</math>.
+We can substitute this into <math>(1)</math> to find that <math>x= \frac{11753}{219} = \frac{161}{3}</math> and <math>m+n = 164</math>.
+<b>Remark:</b> One can come up with the equations in <math>(1)</math> without directly resorting to trig. From similar triangles,
+<math>h/r = 70/x</math>  and <math>h/r = 50/ (92-x)</math>. This implies that <math>70/x =50/(92-x)</math>, so <math>x = 161/3</math>.
+== Solution 3 ==
+From <math>(1)</math> above, <math>x = \frac{70r}{h}</math> and <math>92-x = \frac{50r}{h}</math>. Adding these equations yields <math>92 = \frac{120r}{h}</math>. Thus, <math>x = \frac{70r}{h} = \frac{7}{12}\cdot\frac{120r}{h} = \frac{7}{12}\cdot92 = \frac{161}{3}</math>, and <math>m+n = \boxed{164}</math>.
+We can use <math>(1)</math> from Solution 1 to find that <math>h/r = 70/x</math>  and <math>h/r = 50/ (92-x)</math>.
+This implies that  <math>70/x =50/(92-x)</math> so <math>x = 161/3</math>
+== Solution 4 ==
+Extend <math>AD</math> and <math>BC</math> to meet at a point <math>X</math>. Since <math>AB</math> and <math>CD</math> are parallel, <math>\triangle XCD ~ \triangle XAB</math>. If <math>AX</math> is further extended to a point <math>A'</math> and <math>XB</math> is extended to a point <math>B'</math> such that <math>A'B'</math> is tangent to circle <math>P</math>, we discover that circle <math>P</math> is the incircle of triangle <math>XA'B'</math>. Then line <math>XP</math> is the angle bisector of <math>\angle AXB</math>. By homothety, <math>P</math> is the intersection of the angle bisector of <math>\triangle XAB</math> with <math>AB</math>. By the angle bisector theorem,
+<cmath>\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*}</cmath>
+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 5==
+The area of the trapezoid is <math>\frac{(19+92)h}{2}</math>, where <math>h</math> is the height of the trapezoid.
+Draw lines <math>CP</math> and <math>BP</math>. We can now find the area of the trapezoid as the sum of the areas of the three triangles <math>BPC</math>, <math>CPD</math>, and <math>PBA</math>.
+<math>[BPC] = \frac{1}{2} \cdot 50 \cdot r</math> (where <math>r</math> is the radius of the tangent circle.)
+<math>[CPD] = \frac{1}{2} \cdot 19 \cdot h</math>
+<math>[PBA] = \frac{1}{2} \cdot 70 \cdot r</math>
+<math>[BPC] + [CPD] + [PBA] = 60r + \frac{19h}{2} = [ABCD] = \frac{(19+92)h}{2}</math>
+<math>60r = 46h</math>
+<math>r = \frac{23h}{30}</math>
+From Solution 1 above, <math>\frac{h}{70} = \frac{r}{x}</math>
+Substituting <math>r = \frac{23h}{30}</math>, we find <math>x = \frac{161}{3}</math>, hence the answer is <math>\boxed{164}</math>.
+==Solution 6==
+As the problem tells, the circle is tangent to both sides <math>AD,BC</math>, we can make it up to a triangle <math>QAB</math> and <math>P</math> must lie on its angular bisector. Then we know that <math>AP:BP=7:5</math>, which makes <math>AP = \frac{7}{12}(AB) = \frac{92\times 7}{12} = \frac{161}{3}</math>. Thus, <math>m + n = 164</math>.
 == See also ==
-* [[1992 AIME Problems]]
+{{AIME box|year=1992|num-b=8|num-a=10}}
+[[Category:Intermediate Geometry Problems]]
+{{MAA Notice}}

1992 AIME (Problems • Answer Key • Resources)
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