Answer Happy

7. (15 pts) You might wonder if anything can be said about squares constructed on the sides of any triangle, not just a right triangle (like we did in class at the end of the area discussion). Many things can be said, and here is one of them. (It's weird.) Given a triangle, let point P be in its interior and drop perpendiculars to the sides of the triangle. Construct squares on the sides of our triangle as shown. Prove that regardless of where P is, the combined area of the shaded squares equals the combined area of the unshaded squares! A B [Hint: Pythagorean Theorem (applied a bunch of times) saves the day here. Thing is, there are no right triangles in the picture. So, construct a few segments!] 5