Pythagorean Theorem Problem

Question

Answer 1

ACB is a right triangle, with the right angle at C. An altitude is dropped from vertex C, forming right angles at angle CDA and angle CDB.

Since all angles in a triangle must sum to 180 degrees, the sum of angle CAD and angle ACD is 90, as is the sum of angle CBD and angle BCD.

a)

Therefore, angle CAD must equal to angle BCD and angle CBD must equal angle ACD.

Thus, we have the angles in triangle ACD and triangle CBD equal.

b)

By the AA-similarity theorem, these triangles must be similar. We can then set up the following proportions:

y/a = a/c

a² = cy

c)

(c-y)/b = b/c

b² = c(c-y) = c² - cy

d) Now find a² + b2

a² + b² = cy + c² - cy

a² + b² = c2