Difference between revisions of "2009 AIME I Problems/Problem 12"

Revision as of 18:14, 21 July 2020

Problem

In right $\triangle ABC$ with hypotenuse $\overline{AB}$ , $AC = 12$ , $BC = 35$ , and $\overline{CD}$ is the altitude to $\overline{AB}$ . Let $\omega$ be the circle having $\overline{CD}$ as a diameter. Let $I$ be a point outside $\triangle ABC$ such that $\overline{AI}$ and $\overline{BI}$ are both tangent to circle $\omega$ . The ratio of the perimeter of $\triangle ABI$ to the length $AB$ can be expressed in the form $\frac {m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Solution 1

Let $O$ be center of the circle and $P$ , $Q$ be the two points of tangent such that $P$ is on $BI$ and $Q$ is on $AI$ . We know that $AD:CD = CD:BD = 12:35$ .

Since the ratios between corresponding lengths of two similar diagrams are equal, we can let $AD = 144, CD = 420$ and $BD = 1225$ . Hence $AQ = 144, BP = 1225, AB = 1369$ and the radius $r = OD = 210$ .

Since we have $\tan OAB = \frac {35}{24}$ and $\tan OBA = \frac{6}{35}$ , we have $\sin {(OAB + OBA)} = \frac {1369}{\sqrt {(1801*1261)}},$ $\cos {(OAB + OBA)} = \frac {630}{\sqrt {(1801*1261)}}$ .

Hence $\sin I = \sin {(2OAB + 2OBA)} = \frac {2*1369*630}{1801*1261}$ . let $IP = IQ = x$ , then we have Area $(IBC)$ = $(2x + 1225*2 + 144*2)*\frac {210}{2}$ = $(x + 144)(x + 1225)* \sin {\frac {I}{2}}$ . Then we get $x + 1369 = \frac {3*1369*(x + 144)(x + 1225)}{1801*1261}$ .

Now the equation looks very complex but we can take a guess here. Assume that $x$ is a rational number (If it's not then the answer to the problem would be irrational which can't be in the form of $\frac {m}{n}$ ) that can be expressed as $\frac {a}{b}$ such that $(a,b) = 1$ . Look at both sides; we can know that $a$ has to be a multiple of $1369$ and not of $3$ and it's reasonable to think that $b$ is divisible by $3$ so that we can cancel out the $3$ on the right side of the equation.

Let's see if $x = \frac {1369}{3}$ fits. Since $\frac {1369}{3} + 1369 = \frac {4*1369}{3}$ , and $\frac {3*1369*(x + 144)(x + 1225)}{1801*1261} = \frac {3*1369* \frac {1801}{3} * \frac {1261*4}{3}} {1801*1261} = \frac {4*1369}{3}$ . Amazingly it fits!

Since we know that $3*1369*144*1225 - 1369*1801*1261 < 0$ , the other solution of this equation is negative which can be ignored. Hence $x = 1369/3$ .

Hence the perimeter is $1225*2 + 144*2 + \frac {1369}{3} *2 = 1369* \frac {8}{3}$ , and $BC$ is $1369$ . Hence $\frac {m}{n} = \frac {8}{3}$ , $m + n = 11$ .

Solution 2

As in Solution $1$ , let $P$ and $Q$ be the intersections of $\omega$ with $BI$ and $AI$ respectively.

First, by pythagorean theorem, $AB = \sqrt{12^2+35^2} = 37$ . Now the area of $ABC$ is $1/2*12*35 = 1/2*37*CD$ , so $CD=\frac{420}{37}$ and the inradius of $\triangle ABI$ is $r=\frac{210}{37}$ .

Now from $\triangle CDB \sim \triangle ACB$ we find that $\frac{BC}{BD} = \frac{AB}{BC}$ so $BD = BC^2/AB = 35^2/37$ and similarly, $AD = 12^2/37$ .

Note $IP=IQ=x$ , $BP=BD$ , and $AQ=AD$ . So we have $AI = 144/37+x$ , $BI = 1225/37+x$ . Now we can compute the area of $\triangle ABI$ in two ways: by heron's formula and by inradius times semiperimeter, which yields

$rs=210/37(37+x) = \sqrt{(37+x)(37-144/37)(37-1225/37)x}$

$210/37(37+x) = 12*35/37 \sqrt{x(37+x)}$

$37+x = 2 \sqrt{x(x+37)}$

$x^2+74x+1369 = 4x^2 + 148x$

$3x^2 + 74x - 1369 = 0$

The quadratic formula now yields $x=37/3$ . Plugging this back in, the perimeter of $ABI$ is $2s=2(37+x)=2(37+37/3) = 37(8/3)$ so the ratio of the perimeter to $AB$ is $8/3$ and our answer is $8+3=\boxed{011}$

Solution 3

As in Solution $2$ , let $P$ and $Q$ be the intersections of $\omega$ with $BI$ and $AI$ respectively.

Recall that the distance from a point outside a circle to that circle is the same along both tangent lines to the circle drawn from the point.

Recall also that the length of the altitude to the hypotenuse of a right-angle triangle is the geometric mean of the two segments into which it cuts the hypotenuse.

Let $x = \overline{AD} = \overline{AQ}$ . Let $y = \overline{BD} = \overline{BP}$ . Let $z = \overline{PI} = \overline{QI}$ . The semi-perimeter of $ABI$ is $x + y + z$ . Since the lengths of the sides of $ABI$ are $x + y$ , $y + z$ and $x + z$ , the square of its area by Heron's formula is $(x+y+z)xyz$ .

The radius $r$ of $\omega$ is $\overline{CD}/2$ . Therefore $r^2 = xy/4$ . As $\omega$ is the in-circle of $ABI$ , the area of $ABI$ is also $r(x+y+z)$ , and so the square area is $r^2(x+y+z)^2$ .

Therefore $(x+y+z)xyz = r^2(x+y+z)^2 = \frac{xy(x+y+z)^2}{4}$ Dividing both sides by $xy(x+y+z)/4$ we get: $4z = (x+y+z),$ and so $z = (x+y)/3$ . The semi-perimeter of $ABI$ is therefore $\frac{4}{3}(x+y)$ and the whole perimeter is $\frac{8}{3}(x+y)$ . Now $x + y = \overline{AB}$ , so the ratio of the perimeter of $ABI$ to the hypotenuse $\overline{AB}$ is $8/3$ and our answer is $8+3=\boxed{011}$

Solution 4

We shall yet again let $P$ and $Q$ be the intersections of $AI$ and $BI$ to $\omega$ , respectively. We want to find the perimeter of $ABI$ , which is $AD+BD+BQ+QI+IP+PA$ . We can easily find $AD$ and $BD$ using the fact that $ABC$ , $ACD$ , and $BCD$ are all similar triangles. We get $AD=\frac{144}{37}$ and $\frac{1225}{37}$ . Since $AP$ and $AD$ are tangents to $\omega$ , $AP=AD=\frac{144}{37}$ , and similarly $BQ=BD=\frac{1225}{37}$ . We now wish to find $IP$ and $IQ$ .

Let the center of the given circle be $O$ . We know that $\angle AOP=\angle AOD$ , $\angle BOQ=\angle BOD$ , and $\angle IOQ=\angle IOP$ . Since all six angles sum to $360^{\circ}$ , $\angle AOP+\angle BOQ+\angle IOP=180^{\circ}$ . If we knew the radius of circle $\omega$ now, then we could find $\tan{\angle AOP}$ and $\tan{\angle BOQ}$ , and then we can use the sum (or difference) of tangents formula to find $\tan{\angle IOP}$ , which reveals $IP$ . This means we should find the radius of $\omega$ . We can easily see that the height of triangle $ABC$ from $C$ has length $\frac{12*35}{37}$ , so the radius of $\omega$ is $\frac{210}{37}$ . Now we shall proceed with the above plan.

$\tan{\angle AOP}=\frac{144}{210}$ . $\tan{\angle BOQ}=\frac{1225}{210}$ .

$\tan{\angle IOP}=\tan{(180^{\circ}-\angle AOP-\angle BOQ)}=-\tan{(\angle AOP+\angle BOQ)}$

$=-\frac{\frac{144}{210}+\frac{1225}{210}}{1-\frac{144}{210}*\frac{1225}{210}}=-\frac{1369}{210-\frac{144*1225}{210}}=\frac{1369}{\frac{144*1225}{210}-210}=\frac{37*37}{35*18}$ .

Therefore $OP=\frac{210}{37}\tan{\angle IOP}=\frac{37}{3}$ , and the perimeter of $AIB$ is $2*\frac{37}{3}+2*\frac{144}{37}+2*\frac{1225}{37}=37*\frac{8}{3}$ . Since $AB=37$ , the desired ratio is $\frac{8}{3}$ , and $8+3=\boxed{011}$ .

@@ Line 71: / Line 71: @@
 <math>=-\frac{\frac{144}{210}+\frac{1225}{210}}{1-\frac{144}{210}*\frac{1225}{210}}=-\frac{1369}{210-\frac{144*1225}{210}}=\frac{1369}{\frac{144*1225}{210}-210}=\frac{37*37}{35*18}</math>.
-Therefore <math>IP=\frac{210}{37}\tan{\angle IOP}=\frac{37}{3}</math>, and the perimeter of <math>AIB</math> is <math>2*\frac{37}{3}+2*\frac{144}{37}+2*\frac{1225}{37}=37*\frac{8}{3}</math>. Since <math>AB=37</math>, the desired ratio is <math>\frac{8}{3}</math>, and <math>8+3=\boxed{011}</math>.
+Therefore <math>OP=\frac{210}{37}\tan{\angle IOP}=\frac{37}{3}</math>, and the perimeter of <math>AIB</math> is <math>2*\frac{37}{3}+2*\frac{144}{37}+2*\frac{1225}{37}=37*\frac{8}{3}</math>. Since <math>AB=37</math>, the desired ratio is <math>\frac{8}{3}</math>, and <math>8+3=\boxed{011}</math>.
 == See also ==
 {{AIME box|year=2009|n=I|num-b=11|num-a=13}}
 {{MAA Notice}}

2009 AIME I (Problems • Answer Key • Resources)
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