Special Right Triangles

Special Right Triangles

The special right triangle is a right triangle whose sides are in a particular ratio. According to the special right triangle rules, it is not necessary to use the Pythagorean theorem to get the size of a side.

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A step-by-step guide to solving Special Right Triangles

  • Two special right triangles are \(45^{\circ}-45^{\circ}-90^{\circ}\) and \(30^{\circ}-60^{\circ}-90^{\circ}\) triangles.
  • In a special \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the three angles are \(45^{\circ}, 45^{\circ}\) and \(90^{\circ}\). The lengths of the sides of this triangle are in the ratio of \(1:1:\sqrt{2}\).
  • In a special triangle \(30^{\circ}-60^{\circ}-90^{\circ}\), the three angles are \(30^{\circ}-60^{\circ}-90^{\circ}\). The lengths of this triangle are in the ratio of \(1:\sqrt{3}:2\).

Special Right Triangles – Example 1:

Find the length of the hypotenuse of a right triangle if the length of the other two sides are both 4 inches.


This is a right triangle with two equal sides. Therefore, it must be a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle. Two equivalent sides are 4 inches. The ratio of sides: \(x:x:x\sqrt{2}\). The length of the hypotenuse is \(4\sqrt{2}\) inches. \(x:x:x\sqrt{2}→4:4:4\sqrt{2}\)

Special Right Triangles – Example 2:

The length of the hypotenuse of a triangle is 6 inches. What are the lengths of the other two sides if one angle of the triangle is \(30^{\circ}\)?


The hypotenuse is 6 inches and the triangle is a \(30^°-60^°-90^°\) triangle. Then, one side of the triangle is 3 (it’s half the side of the hypotenuse) and the other side is \(3\sqrt{3}\). (it’s the smallest side times \(\sqrt{3}\)) \(x:x\sqrt{3}:2x→x=3→x:x\sqrt{3}:2x=3:3\sqrt{3}:6\)

Exercises for Special Right Triangles

Find the value of x and y in each triangle.







This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{x=1,y=\frac{\sqrt{3}}{2}}\)
  2. \(\color{blue}{x=4\sqrt{3},y=8\sqrt{3}}\)
  3. \(\color{blue}{x=8\sqrt{2}}\)
  4. \(\color{blue}{x=3\sqrt{2},y=3\sqrt{2}}\)
  5. \(\color{blue}{x=3\sqrt{3},y=6\sqrt{3}}\)
  6. \(\color{blue}{x=4\,y=2\sqrt{2}}\)

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Math and Critical Thinking Challenges: For the Middle and High School Student