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All courses Geometry · G-SRT.8.1 39 of 57
Derive and use trig ratios for 30-60-90 and 45-45-90 special right triangles.

Find separate leg and hypotenuse fields in a45-45-90 triangle

Problem
A \(45-45-90\) triangle has \(\text{one}~\text{leg}\) of length \(x\). Use the \(1:1:\sqrt{2}\) side ratio to express the other leg and hypotenuse in terms of \(x\).
Prompt diagram of a 45-45-90 right triangle with one leg labeled x. The other leg and hypotenuse are marked unknown; their values are withheld. Open full size
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Hint

Identify whether the given is a leg or the hypotenuse.

A45-45-90 triangle has equal legs x and hypotenuse x√2.

Solution walkthrough

01

Identify the special family

\[\text{angles}=45^{\circ},45^{\circ},90^{\circ}\]

The equal acute angles make the triangle isosceles, so its two legs are equal.

02

Use the defining side ratio

\[\text{leg}:\text{leg}:\text{hypotenuse}=1:1:\sqrt{2}\]

A 45-45-90 triangle's hypotenuse is sqrt(2) times either leg.

03

Scale the ratio by x

\[x(1:1:\sqrt{2})=x:x:x~\sqrt{2}\]

The supplied leg x sets the common scale factor to x.

04

Assign the requested sides and check

\[\text{other}~\text{leg}=x;~\text{hypotenuse}=x~\sqrt{2};~x^2+x^2=(x~\sqrt{2})^2\]

The symbolic lengths satisfy the Pythagorean theorem and preserve the special-triangle ratio.

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Another way

  1. Use equal legs x and x, then the Pythagorean theorem gives c=sqrt(x²+x²)=x sqrt(2) for positive x.

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Common mistake

Do not multiply the second leg by sqrt(2). The legs are equal; only the hypotenuse gets the sqrt(2) factor.

Answer diagram of a 45-45-90 triangle using the side ratio 1 to 1 to square root 2: if one leg is x, the other leg is x and the hypotenuse is x square root 2. The equal-leg and Pythagorean checks agree.
The actual isosceles right triangle locates the given and both exact missing side fields.

Solution walkthrough video