ALGEBRA 1

Lesson 10-2 Warm-Up

ALGEBRA 1

“The Pythagorean Theorem” (10-2)

What are the parts of a right triangle?

What is the Pythagorean Theorem?

hypotenuse: The side opposite of the right angle (the longest side of a right triangle)

legs: the sides that form the right angle (the shortest sides of the triangle)

Pythagorean Theorem: In any right triangle, the sum of the squares of the length of the legs, a and b, is equal to the square of the length of the hypotenuse, c.

Note: The legs terms, “a” and “b” are interchangeable (can be switched) since you’re adding them together (Commutative Property), but the hypotenuse is always the “c” term.

ALGEBRA 1

What is the length of the hypotenuse of this triangle?

a2 + b2 = c2 Use the Pythagorean Theorem.

82 + 152 = c2 Substitute 8 for a and 15 for b.

64 + 225 = c2 Simplify.

17 = c Simplify.

The length of the hypotenuse is 17 m.

289 = c2 Find the principal square root of each side.

The Pythagorean TheoremLESSON 10-2

Additional Examples

ALGEBRA 1

A toy fire truck is positioned so that the base of the ladder is 13 cm from the wall. The ladder is extended 28 cm to the wall. How high above the table is the top of the ladder?

Define: Let b = height (in cm) of the ladder from a point 9 cm above the table.

Words: The triangle formed is a right triangle. Use the Pythagorean Theorem.

The Pythagorean TheoremLESSON 10-2

Additional Examples

ALGEBRA 1

Translate: a2 + b2 = c2

132 + b2 = 282 Substitute.169 + b2 = 784 Simplify.

b2 = 615 Subtract 169 from each side.

b2 = 615 Find the principal square root of each side.

b 24.8 Use a calculator and round to the nearest tenth.

(continued)

The height to the top of the ladder is 9 cm higher than 24.8 cm, so it is about 33.8 cm from the table.

The Pythagorean TheoremLESSON 10-2

Additional Examples

ALGEBRA 1

“The Pythagorean Theorem” (10-2)

What is the “Converse (Opposite) of the Pythagorean Theorem”?

Converse of the Pythagorean Theorem: If a triangle has sides a, b, and c, and a2 + b2 = c2 (in other words, if the sum of the squares of the shorter sides equal the square of the longest side), then the triangle is a right triangle with hypotenuse c.

The Converse (Opposite) of the Pythagorean Theorem can be use to determine whether or not a triangle is a right triangle. If the Pythagorean Theorem is a true statement after substituting the lengths of the sides, the triangle is a right triangle

Example: Determine whether the given lengths can form a right triangle.

5 in., 12 in., and 13 in. 7 in., 9 in., and 12 in.

Make a and b the two shorter sides.

Use the Converse of the Pythagorean Theorem.

The triangle is a right triangle.

The triangle is not a right triangle.

ALGEBRA 1

Determine whether the given lengths can be sides of a

right triangle.

a. 5 in., 5 in., and 7 in.

This triangle is not a right triangle.

b. 10 cm, 24 cm, and 26 cm

This triangle is a right triangle.

100 + 576 676 Simplify. 676 = 676

102 + 242 262 Determine whether a2 + b2 = c2, where c is the longest side.

52 + 52 72 Determine whether a2 + b2 = c2, where c is the longest side.

25 + 25 49 Simplify. 50 = 49/

The Pythagorean TheoremLESSON 10-2

Additional Examples

ALGEBRA 1

If two forces pull at right angles to each other, the

resultant force is represented as the diagonal of a rectangle, as

shown in the diagram. The diagonal forms a right triangle with two

of the perpendicular sides of the rectangle.

For a 50–lb force and a 120–lb force, the resultant force is 130 lb. Are the forces pulling at right angles to each other?

16,900 = 16,900

The forces of 50 lb and 120 lb are pulling at right angles to each other.

502 + 1202 1302 Determine whether a2 + b2 = c2 where c is the greatest force.

2500 + 14,400 16,900

The Pythagorean TheoremLESSON 10-2

Additional Examples

ALGEBRA 1

1. Find the missing length 2. Find the missing lengthto the nearest tenth. to the nearest tenth.

3. A triangle has sides of lengths12 in., 14 in., and 16 in. Is thetriangle a right triangle?

4. A triangular flag is attached to a post. The bottom of the flag is 48 in.above the ground. How far from the ground is the top of the flag?

16.6 5.7

no

57 in.

The Pythagorean TheoremLESSON 10-2

Lesson Quiz