Section 8-2 The Pythagorean Theorem

Objectives:• Solve problems using the Pythagorean Theorem

Right Angle: • angle that forms 90°

Hypotenuse: • in a right triangle, it is the side opposite the 90° angle

Leg: • each of the sides forming the right triangle

Pythagorean Theorem: • describes the relationship of the lengths of sides of a

right triangle.

In any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. . . a2 + b2 = c2

a(leg)

b (leg)

c (hypotenuse)

90°This is a right triangle

This is NOT a right triangle

The Pythagorean Theorem

a and b are the legs of the right triangle

c is the hypotenuse and is ALWAYS the longest segment

A Pythagorean Triple

A set of nonzero whole numbers a, b, and c that satisfy the Pythagorean Theorem formula.

If you multiply each number in a Pythagorean Triple by a whole number, the resulting numbers will also form a Triple.

Common Pythagorean Triples

3, 4, 5 5, 12, 13 8, 15, 17

What is the length of the

hypotenuse of this triangle?

The Pythagorean Theorem

Step 1. Write the formula for the

Pythagorean Theorem.

Step 2. Substitute in known values.

Step 3. Solve for the unknown variable.

A television screen measures approximately 15.5 in. high and 19.5 in. wide. A television is advertised by giving the approximate length of the diagonal of its screen. How should this television be advertised?

Real-world Pythagorean Theorem

Draw and label a diagram to model the problem.

15.5

19.5

c

Solve for c, using the Pythagorean Theorem.

The television should be advertised as a 25”.

A toy fire truck is near a toy building on a table such

that the Base of the ladder is 13 cm from the building.

The ladder is extended 28 cm to the building. How high

above the table is the top of the ladder?

Real-world Pythagorean Theorem

Draw a diagram to model the problem.

Solve using the Pythagorean Theorem

The truck is approximately (24.8 + 9) 33.8 cm above the table.

Determine whether the given lengths are sides of a right

triangle.

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

b. 10 cm, 24 cm, and 26 cm

If a triangle has sides of length a and b, and a2 + b2 = c2, then the triangle is a right triangle with hypotenuse of length c. Remember, c is the longest side in a right triangle.

Converse of the Pythagorean Theorem

Not a right triangle

This is a right triangle

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?

Physics and the Pythagorean Theorem

Yes, the forces are pulling at right angles to each other.

c = 16,200 Take the square root.

c 127.27922 Use a calculator.

a2 + b2 = c2 Use the Pythagorean Theorem.902 + 902 = c2 Substitute 90 for a and for b.8100 + 8100 = c2 Simplify.

16,200 = c2

The distance to home plate from second base is about 127 ft.

Use the information to draw a baseball diamond.

A baseball diamond is a square with 90-ft sides. Home

plate and second base are at opposite vertices of the

square. About how far is home plate from second base?

Real-world Connection and the Pythagorean Theorem