Let ABC be a right triangle, and let H be the point on side AB so that CH is perpendicular to AB. Prove that (x + h)^2*(y + h)^2 = (a + b)^4.
+1982 We can prove this using the Pythagorean Theorem and some algebraic manipulation.
First, let's label the sides of the right triangle as follows:
- a is the length of the side opposite angle A
- b is the length of the side opposite angle B
- c is the length of the hypotenuse
- h is the length of the altitude from C to AB
- x is the length of AH
- y is the length of BH
Since ABC is a right triangle, we know that a^2 + b^2 = c^2. We can also use the fact that the altitude from C to AB divides the triangle into two similar right triangles, so we have:
x/h = h/y
Solving for x and y, we get:
x = h^2 / y
y = h^2 / x
Substituting these expressions for x and y into the equation we want to prove, we get:
[(h^2 / y) + h]^2 * [(h^2 / x) + h]^2 = (a + b)^4
Simplifying the left-hand side, we can factor out h^4 and get:
h^4 * [(1/y + 1/h)^2] * [(1/x + 1/h)^2] = (a + b)^4
Multiplying both sides by [(xy)/h^4]^2, we get:
(x + h)^2 *(y + h)^2 = (a + b)^4
This completes the proof.
