# Special Right Triangles

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Special Right Triangles
Lesson 7-3

Investigation In this investigation, you will discover a relationship between the lengths of the legs and the hypotenuse of an isosceles right triangle. This triangle is also referred to as a right triangle because each of its acute angles measures 45°. Folding a square in half can make one of these triangles.

Find the length of the hypotenuse of each isosceles right triangle
Find the length of the hypotenuse of each isosceles right triangle. Simplify the square root each time to reveal a pattern.

Special Right Triangle Theorem
45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, the hypotenuse is times as long as each leg. Verify….

Finding the Length of a Leg
Find the length of a leg of a 45°-45°-90° triangle with a hypotenuse of length 22.

Example A fence around a square garden has a perimeter of 48 feet. Find the approximate length of the diagonal of this square garden.

Finding the Length of a Leg
The distance from one corner to the opposite corner of a square playground is 96 ft. To the nearest foot, how long is each side of the playground?

Investigation The second special right triangle is the right triangle, which is half of an equilateral triangle. Let’s start by using a little deductive reasoning to reveal a useful relationship in right triangles.

Investigation Triangle ABC is equilateral, and segment CD is an altitude. What are m<A and m<B? What are m<ADC and m<BDC? What are m<ACD and m<BCD? Is ΔADC = ΔBDC? Why? Is AD=BD? Why? ~

Find the length of the indicated side in each right triangle by using the conjecture you just made.

Now use the previous conjecture and the Pythagorean formula to find the length of each indicated side.

Special Right Triangle Theorem
30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg.

Finding the Length of the Legs
Find the lengths of the legs of a 30°-60°-90° triangle with hypotenuse of length 8. 60 30 8

Finding the Length of the Legs
Find the lengths of the legs of a 30°-60°-90° triangle with hypotenuse of length 4Ö3. 60 30 4Ö3

Using the Length of a Leg
The longer leg of a 30°-60°-90° triangle has length 18. Find the lengths of the shorter leg and the hypotenuse.

Two Special* Right Triangles

Example What is the area of an equilateral triangle with a side length of 4 cm? 4 cm 4 cm 4 cm

Finding Area

Challenge! Find all side lengths

Classwork P. 369 1-20, odd, 34-40