Triangle Geometry Problem: Proving z²/(x²+y²) = 1/2 Using the Sine Rule

We don’t usually do geometry on Saturdays — but that’s exactly why we should. Here’s a geometry problem to ease into the weekend. Take a look at the triangle in the diagram and enjoy the solution below.

The Problem: A Triangle Geometry Setup

The problem asks us to prove the following identity for a triangle with sides x, y, and z:

z²/(x²+y²) = 1/2

The triangle contains a 45° angle and a right angle (90°), which makes the sine rule the natural tool of choice here.

Applying the Sine Rule to Triangle Geometry

Recall the sine rule: for any triangle, the ratio of each side to the sine of its opposite angle is constant. We’ll apply this to the 45° angle and work our way through the proof in two steps.

Step 1. Using the sine rule with the 45° angle and side y opposite angle A:

z / sin 45° = y / sin A

Solving for sin A:

sin A = y · sin 45° / z = (y√2) / 2z

Squaring both sides:

sin²A = y² / 2z²

Using the Sine Rule with the Complementary Angle

Step 2. The angle opposite side x is (90° − A), since the triangle contains a right angle. Applying the sine rule again:

z / sin 45° = x / sin(90° − A) = x / cos A

Solving for cos A:

cos A = (x√2) / 2z

Squaring:

cos²A = x² / 2z²

Completing the Proof with the Pythagorean Identity

Now we use the fundamental identity sin²A + cos²A = 1. Adding the results from Steps 1 and 2:

y² / 2z² + x² / 2z² = 1

(x² + y²) / 2z² = 1

2z² = x² + y²

z² / (x²+y²) = 1/2 ∎

The elegance here lies in how the sine rule geometry of this triangle maps directly onto the Pythagorean identity — the 45° angle is the key that unlocks the whole thing.