Special Right Triangles 5.5

• Derive the leg lengths of special right triangles.

• Apply the ratios of the legs of special right triangles to find missing information.

Consider a square with sides X.

If we draw in diagonal we’ll obtain two triangles.

ACTake a closer look at the triangle ABC, it’s a Right Triangle!

Applying the Pythagorean Theorem, we obtain the length of our diagonal .AC

Since side lengths are not negative:

Consider an equilateral triangle.

Bisecting angle ACB by drawing a line segment from vertex C to point D on side , we obtain the following:AB

Now, represent the lengths of our equilateral triangle by 2X.

We’ve created a 30°-60°-90° triangle.

We need to determine the length of one of our legs, it’s represented by the ?

Using the Pythagorean Theorem,

Since side lengths are not negative:

x x

x

x

x

x x x x

x

x

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.

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

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.

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.

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.

Find the values of x and y. Give your answers in simplest radical form.

Rationalize the denominator.

Hypotenuse = 2(shorter leg).

Simplify.

y = 2x

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.

Find the values of x and y. Give your answers in simplest radical form.

Simplify.

y = 2(5)

y = 10

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

Find the values of x and y. Give your answers in simplest radical form.

Rationalize the denominator.

Hypotenuse = 2(shorter leg)x = 2y

Simplify.

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.

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.

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

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

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

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

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

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

a.a.

b.b.

c.c. feet1215.605.60,no

feet3.127or290

feet3.127or290

Memorize these formulas.Memorize these formulas.

AssignmentAssignmentDay 1

Mixed Special Right Mixed Special Right TrianglesTriangles

Day 2 30-60-9030-60-9045-45-90 45-45-90

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