8-2 Special Right Triangles

Objective

To use the properties of 45 ° - 45 ° - 90 °  and  30 ° - 60 ° - 90 ° 45 degrees minus 45 degrees minus 90 degrees , and , 30 degrees minus 60 degrees minus 90 degrees triangles

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The Solve It involves triangles with angles 45 ° , 45 ° ,  and  90 ° . 45 degrees comma 45 degrees comma , and , 90 degrees .

Essential Understanding Certain right triangles have properties that allow you to use shortcuts to determine side lengths without using the Pythagorean Theorem.

The acute angles of a right isosceles triangle are both 45 ° 45 degrees angles. Another name for an isosceles right triangle is a 45 ° - 45 ° - 90 ° 45 degrees minus 45 degrees minus 90 degrees triangle. If each leg has length x and the hypotenuse has length y, you can solve for y in terms of x.

x 2 + x 2 = y 2 Use the Pythagorean Theorem . 2 x 2 = y 2 Simplify . x 2 = y Take the positive square root of each side . table with 3 rows and 3 columns , row1 column 1 , x squared , plus , x squared , column 2 equals , y squared , column 3 cap usethecap pythagoreancap theorem . . , row2 column 1 , 2 , x squared , column 2 equals , y squared , column 3 cap simplify , . , row3 column 1 , x square root of 2 , column 2 equals y , column 3 cap takethepositivesquarerootofeachside . . , end table

You have just proved the following theorem.

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