Two Tricky Triangles

Strategies and Hints

  1. The segment from the right angle to the midpoint of the hypotenuse has length 12. This segment (AC in the figure) is the key auxiliary line in the solution.

  2. The altitude BC equals .

  3. The distance AB equals .

  4. Solving 2y2 = 242 for y would give the lengths of the legs of the isosceles right triangle.

Solution

The segment AC has a length of 12. It is also equal to the sum of AB and BC.

Solving the equation = 12 for x will result in x = .

So, k = 12.