Question

A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length a and the other of length b. What is the value of ab?

Answer 1

ab = 1/2.

Explanation:

The sides of the large square have length √5 while the sides of the small one have sides of length 2.

Each corner has a right triangle with legs of length a and b and hypotenuse 2.

So we have the system

a + b = √5

a^2 + b^2 = 2^2 = 4

Using the identity a^2 + b^2 = (a + b)^2 - 2ab:

4 = (√5)^2 - 2ab

4 = 5 - 2ab

2ab = 5 - 4 = 1

ab = 1/2.