special right triangles objectives: 1.to use the properties of 45-45-90 and 30-60-90 right triangles...
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Special Right Triangles
Objectives:
1. To use the properties of 45-45-90 and 30-60-90 right triangles to solve problems
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Investigation 1
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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Investigation 1
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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Investigation 1
Did you notice something interesting about the relationship between the length of the hypotenuse and the length of the legs in each problem of this investigation?
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Special Right Triangle Theorem45°-45°-90° Triangle
Theorem
In a 45°-45°-90° triangle, the hypotenuse is times as long as each leg.
2
2leg hypotenuse
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Example 1
Use deductive reasoning to verify the Isosceles Right Triangle Conjecture.
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Example 2
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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Investigation 2
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 2
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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Investigation 2
Notice that altitude CD divides the equilateral triangle into two right triangles with acute angles that measure 30° and 60°. Look at just one of the 30-60-90 right triangles. How do AC and AD compare?
Conjecture:In a 30°-60°-90° right triangle, if the side
opposite the 30° angle has length x, then the hypotenuse has length -?-.
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Investigation 2
Find the length of the indicated side in each right triangle by using the conjecture you just made.
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Investigation 2
Now use the previous conjecture and the Pythagorean formula to find the length of each indicated side.
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Investigation 2
You should have notice a pattern in your answers. Combine your observations with you latest conjecture and state your next conjecture.
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Special Right Triangle Theorem30°-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.
3
legshorter 2 hypotenuse 3legshorter leglonger
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Two Special Right Triangles
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Example 3
Find the value of each variable. Write your answer in simplest radical form.
1. 2. 3.