# Right Triangles

November 15, 2012 by TMO

A right triangle has one angle equal to 90 degrees. A right triangle can also be an isosceles triangle–which means that it has two sides that are equal. A right isosceles triangle has a 90-degree angle and two 45-degree angles. This is the only right triangle that is an isosceles triangle. This version of the right triangle is so popular that plastic models of them are manufactured and used by architects, engineers, carpenters, and graphic artists in their design and construction work.

Another interesting right triangle is the 30-60-90 degree triangle. The ratio of this triangle’s longest side to its shortest side is “two to one.” That is, the longest side is twice as long as the shortest side. It too is manufactured in plastic and widely used in design, drawing, and building applications.

The Egyptians used this triangle for land surveying. Some believe that they also used it to help design their pyramids. Whether they did or not, the 3-4-5 triangle is still used by surveyors. Carpenters and woodworkers also use it to make their corners square.

Pythagoras was a Greek mathematician who lived about 2500 years ago, and who developed the most famous formula in geometry, possibly in all of mathematics! He proved that, for a right triangle, the sum of the squares of the two sides that join at a right angle equals the square of the third side. The third side–the side opposite the right angle–is called the hypotenuse of the right triangle. The two shorter sides are usually called “legs.”

This formula is called the Pythagorean Theorem in honor of Pythagoras. It is usually written as the equation below, where aand b are the measures of the legs of the triangle and c is the measure of the hypotenuse.

The Pythagorean Theorem has many uses. You can use it to verify whether or not a triangle is a right triangle. Or you can use it to find the missing measures of sides. Let’s use the Pythagorean Theorem to find the missing measure of the leg of the right triangle SAM.

In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. As a rotation, a right angle corresponds to a quarter turn (that is, a quarter of a full circle).

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