# Difference between revisions of "2012 AMC 12B Problems/Problem 25"

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Case 1: <math>A=(0,5)</math>. Then <math>B=(*,0)</math> is impossible. | Case 1: <math>A=(0,5)</math>. Then <math>B=(*,0)</math> is impossible. | ||

− | Case 2: <math>B=(0,5)</math>. Then we look for <math>A=(x,y)</math> such that <math>\angle BAC=90^{\circ}</math> and that <math>C=(*,0)</math>. They are: <math>(A=(x,5), C=(x,0))</math>, <math>(A=(2,4), C=(1,0))</math> and <math>(A=(4,1), C=(3,0))</math>. The product of their values of <math>\tan \angle CBA</math> is <math>5 | + | Case 2: <math>B=(0,5)</math>. Then we look for <math>A=(x,y)</math> such that <math>\angle BAC=90^{\circ}</math> and that <math>C=(*,0)</math>. They are: <math>(A=(x,5), C=(x,0))</math>, <math>(A=(2,4), C=(1,0))</math> and <math>(A=(4,1), C=(3,0))</math>. The product of their values of <math>\tan \angle CBA</math> is <math>\frac{5}{1}\cdot \frac{5}{2} \cdot \frac{5}{3} \cdot \frac{5}{4} \cdot \frac{1}{4} \cdot \frac{2}{3} = 625/144</math>. |

Case 3: <math>C=(0,5)</math>. Then <math>A=(*,0)</math> is impossible. | Case 3: <math>C=(0,5)</math>. Then <math>A=(*,0)</math> is impossible. | ||

Therefore <math>\framebox{B}</math> is the answer. | Therefore <math>\framebox{B}</math> is the answer. |

## Revision as of 13:20, 6 December 2012

## Problem 25

Let . Let be the set of all right triangles whose vertices are in . For every right triangle with vertices , , and in counter-clockwise order and right angle at , let . What is

## Solution 1

Four points in a rectangular arrangement allow 4 possible right angle triangles. These four congruent triangles will form two pairs of different vertice labelling - two ABC and two ACB. These pairs will multiple to equal 1 due to the fact that

Due to the missing point (0,0) in the grid then in the rectangular arrangement (0,0), (x,0), (x,y), (0,y) there is only one triangle which will not cancel out to zero with a reflected version.

So we need to consider all triangles of the form A(x,y), B(0,y), C(x,0). For these triangle tanB = y/x.
Multiplying them all together gives: 1/1 * 1/2 * 1/3 * 1/4 * 2/1 * 2/2 * 2/3 * 2/4 * 3/1 * 3/2 * 3/3 * 3/4 * 4/1 * 4/2 * 4/3 * 4/4 * 5/1 * 5/2 * 5/3 * 5/4 = 625/24

## Solution 2

Consider reflections. For any right triangle with the right labeling described in the problem, any reflection labeled that way will give us . First we consider the reflection about the line . Only those triangles that have one vertex at do not reflect to a traingle . Within those triangles, consider a reflection about the line . Then only those triangles that have one vertex on the line do not reflect to a triangle . So we only need to look at right triangles that have vertices . There are three cases:

Case 1: . Then is impossible.

Case 2: . Then we look for such that and that . They are: , and . The product of their values of is .

Case 3: . Then is impossible.

Therefore is the answer.