Special Right Triangles Examples

Find the value of d.

This triangle is a Gandalf triangle: the 45-45-90 one. Since we know one angle is 45° and another is 90°, we automatically know the third missing angle is also 45° 'cause all three angles need to add up to 180°.

Now we can either use the Pythagorean Theorem or the formula we derived from it. We should know where the formula comes from, so we'll re-derive it real quick.

a² + b² = c²

Since a = b in our case, we'll replace both with x.

x² + x² = c²

2x² = c²

Just square root both sides, and presto.

Awesomesauce. Looking at our cut-in-half box, we know the length of one side. That would be x, and the hypotenuse is d.

Multiply and solve.

c = 3 × 2 = 6

Done and done.

Find the value of x.

We can use our knowledge of the Triforce triangle to find the value of the hypotenuse x. If we aren't sure where to start, the Pythagorean Theorem is still our best friend.

a² + b² = c²

It helps to imagine a Triforce instead of the plain old equilateral triangle. That way, we can see that the hypotenuse of the triangle is twice the length of the shortest leg. Algebraically, that translates to c = 2a.

a² + b² = (2a)²

Isolating and solving for the long leg, b, gives us:

a² + b² = 4a²
b² = 3a²

We now have relationships between all the sides of the triangle. The short leg is a, the hypotenuse is 2a, and the long leg is a√3. We know the long leg is 3, so we can start by finding a.

Nice. Now, we can use that value to solve for the hypotenuse, x.

x = 2a

That's all there is to it.

Find the value of y.

For this one, we have to use our knowledge of both the magical right triangles. Gandalf and the Triforce unite, creating The Legend of the Lord Zelda's Ring. An Academy Award winner for sure.

The only information we have is about the Gandalf triangle, so that's the best place to start. We know the formulas pretty well by now.

The hypotenuse, d, is .

x = 6

We're off to a great start but where do we go from here? Both legs of the Gandalf triangle are equal to each other. Here's where Gandalf and the Triforce combine. The leg of the Gandalf triangle equals the length of the Triforce triangle's long leg. If we remember our Triforce triangle formulas, then it looks like this.

Since we just figured out that x = 6, that deserves some recognition.

Now, we can solve for a, which will give us the length of the Triforce triangle's short side.

No one likes having square roots in the denominator. Luckily, we can fix that right up.

All that's left to do is to combine the sides of the triangles we need. Our value y is the sum of x (the leg of the Gandalf triangle), and a (the short leg of the Triforce triangle).

y = a + x

y ≈ 9.5

We might as well have a fork sticking out of us, because we're done.