Special Right Triangles What are Special Right Triangles
- Slides: 18
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? 3600 • So each angle is? 900
• 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. 450 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 45 -45 -90 triangle.
Standard: MM 2 G 1 b Determine the lengths of the sides of 45 -45 -90 Triangles • Essential Question: What patterns can I use to find the lengths of the sides of a right triangle?
Parts of a 45 -45 -90 Right Triangle Hypotenuse Leg The legs of a 45 -45 -90 are congruent!!!!
Performance Task • Complete Performance Task
45º - 90º Theorems • IN A 45 -45 -90 Δ THE HYPOTENUSE IS TIMES AS LONG AS EACH LEG x 45 o __ __ x 45 o
Example • Find BC and AB A • BC and AC are equal, so BC = 10. 10 • AB is the Hypotenuse and is C 10 B • AB is times AC.
Ex: find x x=5 5 x __ 45 5 45 12
EXAMPLE 1 o o o Find hypotenuse length in a 45 -45 -90 triangle Find the length of the hypotenuse. a. SOLUTION a. By the Triangle Sum Theorem, the measure of the third angle must be 45º. Then the triangle is a 45º-45º 90º triangle, so by Theorem 7. 8, the hypotenuse is 2 times as long as each leg. hypotenuse = leg =8 2 2 o o o 45 -45 -90 Triangle Theorem Substitute.
EXAMPLE 2 o o o Find hypotenuse length in a 45 -45 -90 triangle Find the length of the hypotenuse. b. By the Base Angles Theorem and the Corollary to o the Triangle Sum Theorem, the triangle is a 45 triangle. - 45 - 90 hypotenuse = leg =3 2 =6 o 2 o o 45 -45 -90 Triangle Theorem 2 Substitute. Product of square roots Simplify.
EXAMPLE 3 o o o Find leg lengths in a 45 -45 -90 triangle Find the lengths of the legs in the triangle. SOLUTION By the Base Angles Theorem and the Corollaryoto the o o - 90 Triangle Sum Theorem, the triangle is a 45 - 45 triangle. hypotenuse = leg 5 2 =x 2 2 x 2 5 2 = 2 2 5=x o o o 45 -45 -90 Triangle Theorem Substitute. Divide each side by Simplify. 2
EXAMPLE 4 Standardized Test Practice SOLUTION By the Corollary to the Triangle Sum Theorem, the o o o triangle is a 45 - 90 triangle.
EXAMPLE 4 Standardized Test Practice hypotenuse = leg WX = 25 2 2 The correct answer is B. o o o 45 -45 -90 Triangle Theorem Substitute.
for Examples 1, 2, and 3 GUIDED PRACTICE Find the value of the variable. 1. ANSWER 2. 2 3. ANSWER 2 ANSWER 8 2
GUIDED PRACTICE for Examples 1, 2, and 3 4. Find the leg length of a 45°- 90° triangle with a hypotenuse length of 6. ANSWER 3 2
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