Special Right Triangles
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Transcript of Special Right Triangles
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Special Right Triangles
Lesson 7-3
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Investigation
This triangle is also referred to as a 45-45-90 right triangle because each of its acute angles measures 45°. Folding a square in half can make one of these triangles.
In this investigation, you will discover a relationship between the lengths of the legs and the hypotenuse of an isosceles right triangle.
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Find the length of the hypotenuse of each isosceles right triangle. Simplify the square root each time to reveal a pattern.
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Special Right Triangle Theorem
45°-45°-90° Triangle 45°-45°-90° Triangle TheoremTheorem
In a 45°-45°-90° triangle, the hypotenuse is times as long as each leg.
Verify….
2
2leg hypotenuse
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Finding the Length of a Leg
Find the length of a leg of a 45°-45°-90° triangle with a hypotenuse of length 22.
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Example
A fence around a square garden has a perimeter of 48 feet. Find the approximate length of the diagonal of this square garden.
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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?
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Investigation
The second special right triangle is the 30-60-90 right triangle, which is half of an equilateral triangle.
Let’s start by using a little deductive reasoning to reveal a useful relationship in 30-60-90 right triangles.
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Investigation
Triangle ABC is equilateral, and segment CD is an altitude.
1. What are m<A and m<B?
2. What are m<ADC and m<BDC?
3. What are m<ACD and m<BCD?
4. Is ΔADC = ΔBDC? Why?
5. Is AD=BD? Why?
~
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Find the length of the indicated side in each right triangle by using the conjecture you just made.
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Now use the previous conjecture and the Pythagorean formula to find the length of each indicated side.
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Special Right Triangle Theorem
30°-60°-90° Triangle 30°-60°-90° Triangle TheoremTheorem
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.
3
legshorter 2 hypotenuse 3legshorter leglonger
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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
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Finding the Length of the Legs Find the lengths of the legs of a 30°-
60°-90° triangle with hypotenuse of length 43.
60
30
43
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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.
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Two Special* Right Triangles
*what’s so special about them?
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ExampleFind the value of each variable. Write
your answer in simplest radical form.
1. 2. 3.
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Example
Find the value of each variable. Write your answer in simplest radical form.
1. 2. 3.
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Example
What is the area of an equilateral triangle with a side length of 4 cm?
4 cm
4 cm 4 cm
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Finding Area
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Challenge! Find all side lengths
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Classwork
P. 3691-20, 21-29 odd, 34-40