special right triangles 5.5. derive the leg lengths of special right triangles. apply the ratios of...
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Special Right Triangles 5.5
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• Derive the leg lengths of special right triangles.
• Apply the ratios of the legs of special right triangles to find missing information.
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Consider a square with sides X.
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If we draw in diagonal we’ll obtain two triangles.
ACTake a closer look at the triangle ABC, it’s a Right Triangle!
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Applying the Pythagorean Theorem, we obtain the length of our diagonal .AC
Since side lengths are not negative:
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Consider an equilateral triangle.
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Bisecting angle ACB by drawing a line segment from vertex C to point D on side , we obtain the following:AB
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Now, represent the lengths of our equilateral triangle by 2X.
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We’ve created a 30°-60°-90° triangle.
We need to determine the length of one of our legs, it’s represented by the ?
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Using the Pythagorean Theorem,
Since side lengths are not negative:
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x x
x
x
x
x x x x
x
x
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Finding Side Lengths in a 45°- 45º- 90º Triangle
Find the value of x. Give your answer in simplest radical form.
By the Triangle Sum Theorem, the measure of the third angle in the triangle is 45°. So it is a 45°-45°-90° triangle with a leg length of 8.
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Find the value of x. Give your answer in simplest radical form.
x = 20 Simplify.
By the Triangle Sum Theorem, the measure of the third angle in the triangle is 45°. So it is a 45°-45°-90° triangle with a leg length of
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Find the value of x. Give your answer in simplest radical form.
The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 5.
Rationalize the denominator.
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Find the value of x. Give your answer in simplest radical form.
The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 16.
Rationalize the denominator.
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Find the values of x and y. Give your answers in simplest radical form.
Hypotenuse = 2(shorter leg)22 = 2x
Divide both sides by 2.11 = x
Substitute 11 for x.
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Find the values of x and y. Give your answers in simplest radical form.
Rationalize the denominator.
Hypotenuse = 2(shorter leg).
Simplify.
y = 2x
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Find the values of x and y. Give your answers in simplest radical form.
Hypotenuse = 2(shorter leg)
Divide both sides by 2.
y = 27 Substitute for x.
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Find the values of x and y. Give your answers in simplest radical form.
Simplify.
y = 2(5)
y = 10
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Find the values of x and y. Give your answers in simplest radical form.
Hypotenuse = 2(shorter leg)
Divide both sides by 2.
Substitute 12 for x.
24 = 2x
12 = x
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Find the values of x and y. Give your answers in simplest radical form.
Rationalize the denominator.
Hypotenuse = 2(shorter leg)x = 2y
Simplify.
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An ornamental pin is in the shape of an equilateral triangle. The length of each side is 6 centimeters. Josh will attach the fastener to the back along AB. Will the fastener fit if it is 4 centimeters long?The equilateral triangle is divided into two 30°-60°-90° triangles.
The height of the triangle is the length of the longer leg. Find the length x of the shorter leg.
Hypotenuse = 2(shorter leg)6 = 2x
3 = x Divide both sides by 2.Find the length h of the longer leg.
The pin is approximately 5.2 centimeters high. So the fastener will fit.
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What if…? A manufacturer wants to make a larger clock with a height of 30 centimeters. What is the length of each side of the frame? Round to the nearest tenth.
Step 1 The equilateral triangle is divided into two 30º-60º-90º triangles.
The height of the triangle is the length of the longer leg.
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Step 2 Find the length x of the shorter leg.
Each side is approximately 34.6 cm.
Step 3 Find the length y of the longer leg.
Rationalize the denominator.
y = 2x
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Find the exact answer of the missing side.Find the exact answer of the missing side.
a.a. b.b. c.c.2
29x 20y10x
2
23y
2
6x
d.d. e.e. f.f. 22.3x 25x 36y33x
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Find the exact answer of the missing side.Find the exact answer of the missing side.
a.a. b.b. c.c. 318y18x 28x 29x
d.d. e.e. f.f. 8y11y 5x
x
y
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Find the exact answer of the missing side.Find the exact answer of the missing side.
a.a. b.b. c.c. 30y315x 29y 310y10x
d.d. e.e. f.f. 8y16x 24y24x 37y7x
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a.a. b.b. c.c. 222x 220y20x 28x
d.d. e.e. f.f. 21x 32s 6x
Find the exact answer of the missing side.Find the exact answer of the missing side.
x x
60
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a.a. b.b. c.c.24d312c
12b12a
34y32x
39d27c
69b39a
d.d. e.e. f.f.3530b
15a
312c
36b18a
220c
20b210a
Find the exact answer of the missing side.Find the exact answer of the missing side.
xy
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a.a.
b.b.
c.c. feet1215.605.60,no
feet3.127or290
feet3.127or290
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Memorize these formulas.Memorize these formulas.
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AssignmentAssignmentDay 1
Mixed Special Right Mixed Special Right TrianglesTriangles
Day 2 30-60-9030-60-9045-45-90 45-45-90
FrontFront