|
|
| Line 1: |
Line 1: |
| − | ==Solution 3==
| |
| − |
| |
| − | As in Solution <math>2</math>, let <math>P</math> and <math>Q</math> be the intersections of <math>\omega</math> with <math>BI</math> and <math>AI</math> 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 <math>x = \overline{AD} = \overline{AQ}</math>. Let <math>y = \overline{BD} = \overline{BP}</math>. Let <math>z = \overline{PI} = \overline{QI}</math>. The semi-perimeter of <math>ABI</math> is <math>x + y + z</math>.
| |
| − | Since the lengths of the sides of <math>ABI</math> are <math>x + y</math>, <math>y + z</math> and <math>x + z</math>, the square of its area by Heron's formula is <math>(x+y+z)xyz</math>.
| |
| − |
| |
| − | The radius <math>r</math> of <math>\omega</math> is <math>\overline{CD}/2</math>. Therefore <math>r^2 = xy/4</math>. As <math>\omega</math> is the in-circle of <math>ABI</math>, the area of <math>ABI</math> is also <math>r(x+y+z)</math>, and so the square area is <math>r^2(x+y+z)^2</math>.
| |
| − |
| |
| − | Therefore <cmath>(x+y+z)xyz = r^2(x+y+z)^2 = \frac{xy(x+y+z)^2}{4}</cmath> Dividing both sides by <math>xy(x+y+z)/4</math> we get: <cmath>4z = (x+y+z),</cmath> and so <math>z = (x+y)/3</math>. The semi-perimeter of <math>ABI</math> is therefore <math>\frac{4}{3}(x+y)</math> and the whole perimeter is <math>\frac{8}{3}(x+y)</math>. Now <math>x + y = \overline{AB}</math>, so the ratio of the perimeter of <math>ABI</math> to the hypotenuse <math>\overline{AB}</math> is <math>8/3</math> and our answer is <math>8+3=\boxed{011}</math>
| |
| − |
| |
| − |
| |
| − | and <math>8+3=\boxed{011}</math>.
| |
| − |
| |
| | == See also == | | == See also == |
| | {{AIME box|year=2009|n=I|num-b=11|num-a=13}} | | {{AIME box|year=2009|n=I|num-b=11|num-a=13}} |
| | [[Category: Intermediate Geometry Problems]] | | [[Category: Intermediate Geometry Problems]] |
| | {{MAA Notice}} | | {{MAA Notice}} |
Revision as of 05:27, 30 September 2022
