Special Right Triangles. What are Special Right Triangles? There are 2 types of Right triangles that are considered special. We will talk about only one.

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Presentation transcript:

Special Right Triangles

What are Special Right Triangles? There are 2 types of Right triangles that are considered special. We will talk about only one of these today.

Imagine a square. What makes a square…square? All 4 sides congruent All 4 angles congruent What do the 4 angles add up to? So each angle is? 90 0

Now divide this square with a diagonal. What kind of triangles are formed?? 2 congruent sides and a right angle in each triangle. ISOSCELES RIGHT TRIANGLES!!!

Each triangle has a right angle and 2 congruent sides. If a triangle has 2 congruent sides, then it must have 2 congruent angles. What are the 2 angles? This is the first special right triangle. A triangle. 45 0

Standard: MM2G1b Determine the lengths of the sides of Triangles Essential Question: What patterns can I use to find the lengths of the sides of a right triangle?

Parts of a Right Triangle Hypotenuse Leg The legs of a are congruent!!!!

Performance Task Complete Performance Task

45º - 45º - 90º Theorems IN A Δ THE HYPOTENUSE IS TIMES AS LONG AS EACH LEG 45 o x x __

Example 10 C A B Find BC and AB 10 BC and AC are equal, so BC = 10. AB is the Hypotenuse and is times AC. AB is

Ex: find x 5 5 x 45 __ x= x

EXAMPLE 1 Find hypotenuse length in a triangle o o o Find the length of the hypotenuse. a. SOLUTION hypotenuse = leg 2 = 8= 8 2 Substitute Triangle Theorem o o o By the Triangle Sum Theorem, the measure of the third angle must be 45 º. Then the triangle is a 45 º -45 º - 90 º triangle, so by Theorem 7.8, the hypotenuse is 2 times as long as each leg. a.

EXAMPLE 2 Find hypotenuse length in a triangle o o o hypotenuse = leg 2 Substitute Triangle Theorem o o o = 3 22 = 3 2 Product of square roots = 6 Simplify. b. By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle o o o Find the length of the hypotenuse. b.

EXAMPLE 3 Find leg lengths in a triangle o o o Find the lengths of the legs in the triangle. SOLUTION By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a triangle o o o hypotenuse = leg 2 Substitute Triangle Theorem o o o 2 5 = x= x = 2 x 2 5 = x Divide each side by 2 Simplify.

EXAMPLE 4 Standardized Test Practice SOLUTION By the Corollary to the Triangle Sum Theorem, the triangle is a triangle o o o

EXAMPLE 4 Standardized Test Practice hypotenuse = leg 2 Substitute Triangle Theorem o o o = 252 WX The correct answer is B.

GUIDED PRACTICE for Examples 1, 2, and 3 Find the value of the variable ANSWER

GUIDED PRACTICE for Examples 1, 2, and 3 4. Find the leg length of a 45°- 45°- 90° triangle with a hypotenuse length of ANSWER