Special right triangles

Learning Objectives Identify 45-45-90 and 30-60-90 triangles from given information. Recall and apply the side-length ratio theorems for both 45-45-90 and 30-60-90 triangles. Solve for all unknown side lengths of a special right triangle when given a single side length. Calculate the exact values of sine, cosine, and tangent for 30°, 45°, and 60° angles without a calculator. Connect the side ratios of special right triangles to the coordinates of key points on the unit circle. Solve multi-step geometric problems that involve special right triangles. Ever wondered how architects design perfect trusses or how video game developers calculate precise movements? 📐 It often boils down to two surprisingly simple, 'special' triangles! This tutorial explores the propertie...

TermDefinitionExample 45-45-90 TriangleAn isosceles right triangle where the two acute angles are both 45 degrees. The two legs are always equal in length.A square cut in half along its diagonal forms two 45-45-90 triangles. 30-60-90 TriangleA right triangle with acute angles of 30 and 60 degrees. It is a scalene triangle, meaning all three sides have different lengths.An equilateral triangle cut in half by an altitude forms two 30-60-90 triangles. LegsThe two sides of a right triangle that form the 90-degree angle.In a triangle with sides a, b, and c, if angle C is 90°, then sides a and b are the legs. HypotenuseThe longest side of a right triangle, located opposite the 90-degree angle.In a triangle with sides a, b, and c, if angle C is 90°, then side c is the hypotenuse. Exact ValueA ma...

45-45-90 Triangle Theorem If the length of a leg is `x`, then the other leg is also `x`, and the hypotenuse is `x\sqrt{2}`. Ratio of sides: `x : x : x\sqrt{2}`. Use this rule when you have a right triangle with two 45-degree angles. If you know one leg, you can find the other leg and the hypotenuse. If you know the hypotenuse, you can find the legs by dividing by \sqrt{2}. 30-60-90 Triangle Theorem If the short leg (opposite the 30° angle) is `x`, then the long leg (opposite the 60° angle) is `x\sqrt{3}`, and the hypotenuse (opposite the 90° angle) is `2x`. Ratio of sides: `x : x\sqrt{3} : 2x`. Use this rule for 30-60-90 triangles. The key is to always find the length of the short leg (`x`) first, as the other two sides are defined in terms of it.