# Special

## Right Triangles

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Activity: The Isosceles (45-45) Right Triangle. An Isosceles right triangle is a special triangle with several special properties. Thes properties result in shortcuts that make it easy to find unknown measures of parts of an isosceles right triangles

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Activity: The 30-60-90 Right Triangle. The right triangle, which is half of an equilateral (equiangular) triangle, has special properties also.

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Isosceles Right ()Triangle Theorem: In an isosceles right () triangle, the hypotenuse is times as long as a leg.

Triangle Theorem: In a triangle, the hypotenuse is twice as long as the shorter leg and the longer leg is times as long as the shorter leg.

In the triangle to the right we can apply the Pythagorean thereom to the hypotenuse, s :

So we see that the sides in a triangle are .

Examples of using the Isosceles Right Triangle and Theorem:

Find x:

1.Since we have an isosceles right () triangle, the legs are equal, so x =12.

2. Since we have an isosceles right () triangle, the hypotenuse is as long as the leg so

x = 12.

Find the Area of the following triangle:

We are given the length of the hypotenuse so we will first use this part of the Theorem which states: the hypotenuse is twice the shorter leg, so the shorter leg (the base of our triangle) is half the hypotenuse,or, feet.

The next part of the theorem says longer leg is times as long as the shorter leg. So the altitude of our triangle (the longer leg) is feet. The formula for the area of a triangle is base times altitude, so we can find the area of our triangle is :feet.

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