1989 AIME Problems/Problem 10

Problem

Let $a_{}^{}$ , $b_{}^{}$ , $c_{}^{}$ be the three sides of a triangle, and let $\alpha_{}^{}$ , $\beta_{}^{}$ , $\gamma_{}^{}$ , be the angles opposite them. If $a^2+b^2=1989^{}_{}c^2$ , find

$\frac{\cot \gamma}{\cot \alpha+\cot \beta}$

Solution

We can draw the altitude h to c, to get two right triangles.

$\cot{\alpha}+\cot{\beta}=\frac{c}{h}$ , from the definition of the cotangent.

From the definition of area, $h=\frac{2A}{c}$ , so therefore $\cot{\alpha}+\cot{\beta}=\frac{c^2}{2A}$

Now we evaluate the numerator:

$\cot{\gamma}=\frac{\cos{\gamma}}{\sin{\gamma}}$ .

$\cos{\gamma}=\frac{1988c^2}{2ab}$ , from the Law of Cosines

$\sin{\gamma}=\frac{c}{2R}$ , where R is the circumradius.

$\cot{\gamma}=\frac{1988cR}{ab}$

Since $R=\frac{abc}{4A}$ , $\cot{\gamma}=\frac{1988c^2}{4A}$

$\frac{\cot \gamma}{\cot \alpha+\cot \beta}=\frac{\frac{1988c^2}{4A}}{\frac{c^2}{2A}}=\frac{1988}{2}=994$

Page

Toolbox

Search

1989 AIME Problems/Problem 10

Problem

Solution

See also