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Special right triangles
Description
Exercise Name: Special right triangles
Math Missions: High school geometry Math Mission, Trigonometry Math Mission
Types of Problems: 2

The Special right triangles exercise appears under the High school geometry Math Mission and Trigonometry Math Mission. This exercise practices recognizing and calculating the sides in the two special right triangles.

## Types of Problems

There are two types of problems in this exercise:

1. Find the missing side in the 30-60-90 triangle: This problem provides a picture with a 30-60-90 triangle and with at least one side given. The student is expected to use the known relationships to find the missing side length.
2. Find the missing side in the 45-45-90 triangle: This problem provides a picture with a 45-45-90 triangle and with at least one side given. The student is expected to use the known relationships to find the missing side length.

## Strategies

Knowledge of the ratios of the two special triangles would ensure success on this exercise, although basic triangle facts and the Pythagorean Theorem are also sufficient.

1. In the 30-60-90 triangle if the short side is x then the hypotenuse is ${2x}$ and the longer leg is ${x\sqrt{(3)}}$
2. In the 45-45-90 triangle if the legs are x then the hypotenuse is ${x\sqrt{(2)}}$.
3. Answers are to be provided as exact radicals, not decimal approximations.

## Real-life applications

1. These two triangles are used to find many of the standard positions on the Unit Circle, which can be viewed as the "times-table" or trigonometry.
2. Trigonometry has countless applications in astronomy, architecture, and physics.