For Exercises 1 and 2, find the value of x. Give your answer in simplest radical form.

1. 2.

Simplify each expression.

3. 4.

A set of three nonzero whole numbers a, b, and c such that a2 + b2 = c2 is called a Pythagorean triple.

Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain.

a2 + b2 = c2

142 + 482 = c2 Substitute 14 for a and 48 for b.

Pythagorean Theorem

2500 = c2 Multiply and add.

50 = c Find the positive square root.

The side lengths are nonzero whole numbers that satisfy the equation a2 + b2 = c2, so they form a Pythagorean triple.

The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths.

You can also use side lengths to classify a triangle as acute or obtuse.

A

B

C

c

b

a

To understand why the Pythagorean inequalities are true, consider ∆ABC.

By the Triangle Inequality Theorem, the sum of any two side lengths of a triangle is greaterthan the third side length.

Remember!

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.

5, 7, 10

Step 1 Determine if the measures form a triangle.

By the Triangle Inequality Theorem, 5, 7, and 10 can be the side lengths of a triangle.

Step 2 Classify the triangle.

c2 = a2 + b2? Compare c2 to a2 + b2.

102 = 52 + 72

? Substitute the longest side for c.

100 = 25 + 49

?Multiply.

100 > 74 Add and compare.

Since c2 > a2 + b2, the triangle is obtuse.

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.

5, 8, 17

Step 1 Determine if the measures form a triangle.

Since 5 + 8 = 13 and 13 > 17, these cannot be the side lengths of a triangle.

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.

7, 12, 16

Step 1 Determine if the measures form a triangle.

By the Triangle Inequality Theorem, 7, 12, and 16 can be the side lengths of a triangle.

Step 2 Classify the triangle.

c2 = a2 + b2?

Compare c2 to a2 + b2.

162 = 122 + 72

?Substitute the longest side for c.

256 = 144 + 49?

Multiply.

256 > 193 Add and compare.

Since c2 > a2 + b2, the triangle is obtuse.

Justify and apply properties of 45°-45°-90° triangles.

Justify and apply properties of 30°- 60°- 90° triangles.

A diagonal of a square divides it into two congruent isosceles right triangles. Since the base angles of an isosceles triangle are congruent, the measure of each acute angle is 45°. So another name for an isosceles right triangle is a 45°-45°-90° triangle.

A 45°-45°-90° triangle is one type of special right triangle. You can use the Pythagorean Theorem to find a relationship among the side lengths of a 45°-45°-90° 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.

Rationalize the denominator.

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

x = 20 Simplify.

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.

Jana is cutting a square of material for a tablecloth. The table’s diagonal is 36 inches. She wants the diagonal of the tablecloth to be an extra 10 inches so it will hang over the edges of the table. What size square should Jana cut to make the tablecloth? Round to the nearest inch.

Jana needs a 45°-45°-90° triangle with a hypotenuse of 36 + 10 = 46 inches.

A 30°-60°-90° triangle is another special right triangle. You can use an equilateral triangle to find a relationship between its side lengths.

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

22 = 2x Hypotenuse = 2(shorter leg)

11 = x Divide both sides by 2.

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).

y = 2x

Simplify.

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.

y = 2(5)

Simplify.y = 10

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

Hypotenuse = 2(shorter leg)24 = 2x

Divide both sides by 2.12 = x

Substitute 12 for x.

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?Step 1 The equilateral triangle is divided into

two 30°-60°-90° triangles.

The height of the triangle is the length of the longer leg.

Step 2 Find the length x of the shorter leg.

Hypotenuse = 2(shorter leg)6 = 2x3 = x Divide both sides by 2.

Step 3 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.

Rationalize the denominator.

Step 3 Find the length y of the longer leg.

Hypotenuse = 2(shorter leg)

y = 2x

Simplify.Each side is approximately 34.6 cm.