Special Right Triangles
In short: Special right triangles come in two flavors, 45-45-90 and 30-60-90, whose sides follow fixed multiples, so a missing length comes from simple multiplication.
GeometryTrigonometryG-SRT.C.6
What is Special Right Triangles?
Two special right triangles keep turning up because their sides are locked into tidy ratios. Slice a square along its diagonal and you obtain the 45-45-90 type. Cut an equilateral triangle down its middle and you obtain the 30-60-90 type.
In the first, the two legs match, call them \(x\), and the hypotenuse is \(x\sqrt{2}\). In the second, the side across from \(30^{\circ}\) is \(x\), the side across from \(60^{\circ}\) is \(x\sqrt{3}\), and the hypotenuse is \(2x\). One known side unlocks the other two.
How to spot it
- The angles of a right triangle are 45 and 45, or 30 and 60: the signature of special right triangles.
- A figure is a square with its diagonal drawn, or an equilateral triangle with its height drawn.
- The answer is wanted in exact form with a radical, such as \(7\sqrt{2}\).
- A right triangle has two equal legs, or its hypotenuse is exactly double one leg.
The method
- With special right triangles, first decide which pattern the angles match: 45-45-90 or 30-60-90.
- Name the smallest side \(x\): a leg in the first pattern, the side across from \(30^{\circ}\) in the second.
- List the other sides, \(x\) and \(x\sqrt{2}\), or \(x\sqrt{3}\) and \(2x\), then match the given side to find \(x\).
- Compute the side you need, leaving radicals exact unless a decimal is requested.
Special Right Triangles: solved example
Problem
An equilateral banner has sides of 18 cm. Find its height, which splits the banner into two 30-60-90 triangles.
Solution
The height lands on the midpoint of the base, so each half-banner has hypotenuse 18 cm and short leg \(\frac{18}{2} = 9\) cm. That short leg is \(x = 9\).
The height faces \(60^{\circ}\), so it equals \(x\sqrt{3} = 9\sqrt{3} \approx 15.59\). The banner stands \(9\sqrt{3} \approx 15.6\) cm tall.
Try it
Question 1
A square window has sides of 11 inches. Find the exact length of its diagonal.
Show the answer
The diagonal is the hypotenuse of a 45-45-90 triangle with legs 11: \(11\sqrt{2}\) inches, about 15.56.
Question 2
In a 30-60-90 triangle the longer leg is \(12\sqrt{3}\). Find the other two sides.
Show the answer
The longer leg is \(x\sqrt{3}\), so \(x = 12\), and the hypotenuse is \(2x = 24\).
Special Right Triangles vs. Pythagorean Triple
A Pythagorean triple, such as 5-12-13, is a trio of whole numbers satisfying \(a^2 + b^2 = c^2\), and its angles are not neat. These shapes are defined by their angles, and their sides involve radicals. A 45-45-90 triangle can never have all three sides as whole numbers.
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