Auxiliary similar triangles Pythagorean theorem (direct and inverse).

In triangle $ABC$, altitudes $AA_1$ and $BB_1$ are drawn. Find $AC$, if
a) $AA_1=4, BB_1=5, BC=6$

b) $A_1C=8, B_1C=5, BB_1=12$.
a) The product of the base and height for the given triangle is constant.

b) Triangles $C A A_{1}$ and $C B B_{1}$ are similar.

## Solution

a) From the equality $B C \cdot A A_{1}=A C \cdot B B_{1}$ (twice the area of triangle $A B C$), we find that $A C=24 / 5$.

b) By the Pythagorean theorem, $B C^{2}=B_{1} B^{2}+B_{1} C^{2}=169$.

From the similarity of right triangles $C A A_{1}$ and $C B B_{1}$, it follows that $A C: C A_{1}=B C: C B_{1}$, from which $A C=$ $B C \cdot C A_{1 /} C B_{1}=104 / 5$.

![](https://cdn.mathpix.com/cropped/2024_05_06_14311972f49418f31424g-30.jpg?height=378&width=560&top_left_y=1186&top_left_x=746)

## Answer

a) 4.8;
b) 20.8.