Special right triangles

The  exercise appears under the 8th Grade 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.Srt1.png
 * 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.Srt2.png

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 xsqrt(3)
 * 2) In the 45-45-90 triangle if the legs are x then the hypotenuse is xsqrt(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. Trigonometry has countless applications in astronomy, architecture and physics.