Question

A right triangle has the property that the lengths of its sides form a geometric progression, (i.e. the ratio of shorter leg to the longer leg is the same as the ratio of the longer leg to the hypotenuse.) What is the ratio of the hypotenuse to the shorter leg?

Answer 1

The ratio of the hypotenuse to the shorter leg in a right triangle where the sides form a geometric progression is (1 + √5)/2, which is the square of the golden ratio.

Step-by-step explanation:

The question involves finding the ratio of the sides of a right triangle where the sides form a geometric progression. Let's denote the lengths of the shorter leg, longer leg, and hypotenuse as a, b, and c respectively. Since the sides form a geometric progression, we have b/a = c/b. Let's denote the common ratio by r, meaning b = ar and c = br = ar^2.

Applying the Pythagorean theorem, we get:

a^2 + b^2 = c^2

Substituting the geometric progression terms, we have:

a^2 + (ar)^2 = (ar^2)^2

a^2 + a^2r^2 = a^2r^4

Dividing all terms by a^2 gives us:

1 + r^2 = r^4

Solving for r, we get r^2 = (1 + √5)/2, which is the golden ratio. Therefore, the hypotenuse to the shorter leg ratio, which is c/a, will be r^2. So the ratio of the hypotenuse to the shorter leg is r^2, or (1 + √5)/2.