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45°-45°-90° Triangles

There is a special relationship among the measures of the sides of a 45 ° 45 ° 90 ° triangle.

45 ° 45 ° 90 ° triangle is a commonly encountered right triangle whose sides are in the proportion 1 : 1 : 2 . The measures of the sides are x , x , and x 2 .

Math diagram

In a 45 ° 45 ° 90 ° triangle, the length of the hypotenuse is 2 times the length of a leg.

To see why this is so, note that by the Converse of the Pythagorean Theorem , these values make the triangle a right triangle.

x 2 + x 2 = 2 x 2 = ( x 2 ) 2

 

Note that an isosceles right triangle must be a 45 ° 45 ° 90 ° triangle.

 

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