The Pythagorean Theorem The Pythagorean Theorem and Its Converseand Its Converse

Use the Pythagorean Theorem

Use the Converse of the Pythagorean Theorem

Theorem 7.4In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the legs.

a2 + b2 = c2

Draw a ┴ from C to AB

1. and

REASONS

ca = a

ecb = b

f

3. ce = a2 and cf = b2

4. ce + cf = a2 + b2

5. c(e + f) = a2 + b2

6. e + f = c

7. c2 = a2 + b2

1. Perpendicular Postulate

2. Geometric Mean Theorem

3. Cross Product Property

4. Addition Property

5. Distributive Property

6. Segment Add. Postulate

7. Substitution Property

STATEMENTS

1.

2.

LONGITUDE AND LATITUDE Carson City, Nevada, is located at about 120 degrees longitude and 39 degrees latitude. NASA Ames is located about 122 degrees longitude and 37 degrees latitude. Use the lines of longitude and latitude to find the degree distance to the nearest tenth degree if you were to travel directly from NASA Ames to Carson City.

Use the Pythagorean Theorem to find the distance in degrees from NASA Ames to Carson City, represented by c.

The change in latitude is or 2 degrees latitude. Let this distance be b.

The change in longitude between NASA Ames and Carson City is or 2 degrees. Let this distance be a.

Pythagorean Theorem

Simplify.

Add.

Take the square root of each side.

Use a calculator.

Answer: The degree distance between NASA Ames and Carson City is about 2.8 degrees.

Find d.

Pythagorean Theorem

Simplify.

Subtract 9 from each side.

Take the square root of each side.

Use a calculator.

Answer:

Find x.

Answer:

Theorem 7.5If the square of the length of the longest side of the triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

If c2 = a2 + b2, then ∆ABC is a right triangle.

COORDINATE GEOMETRY Verify that is a right triangle.

Use the Distance Formula to determine the lengths of the sides.

Subtract.

Simplify.

Subtract.

Simplify.

Subtract.

Simplify.

By the converse of the Pythagorean Theorem, if the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle.

Converse of the Pythagorean Theorem

Simplify.

Add.

Answer: Since the sum of the squares of two sides equals the square of the longest side, is a right triangle.

COORDINATE GEOMETRY Verify that is a right triangle.

Answer: is a right triangle because

A Pythagorean Triple is three whole numbers that satisfy the equation a2 + b2 = c2 , where the longest side is the hypotenuse, c, and a and b are the two legs.

i.e. 3, 4, 5

Determine whether 9, 12, and 15 are the sides of a right triangle. Then state whether they form a Pythagorean triple.

Since the measure of the longest side is 15, 15 must be c. Let a and b be 9 and 12.

Pythagorean Theorem

Simplify.

Add.

Answer: These segments form the sides of a right triangle since they satisfy the Pythagorean Theorem. The measures are whole numbers and form a Pythagorean triple.

Determine whether 21, 42, and 54 are the sides of a right triangle. Then state whether they form a Pythagorean triple.

Pythagorean Theorem

Simplify.

Add.

Answer: Since , segments with these measures cannot form a right triangle. Therefore, they do not form a Pythagorean triple.

Pythagorean Theorem

Simplify.

Add.

Determine whether 4, and 8 are the sides of a right triangle. Then state whether they form a Pythagorean triple.

Answer: Since 64 = 64, segments with these measures form a right triangle. However, is not a whole number. Therefore, they do not form a Pythagorean triple.

Answer: The segments form the sides of a right triangle and the measures form a Pythagorean triple.

Answer: The segments do not form the sides of a right triangle, and the measures do not form a Pythagorean triple.

Answer: The segments form the sides of a right triangle, but the measures do not form a Pythagorean triple.

Determine whether each set of measures are the sides of a right triangle. Then state whether they form a Pythagorean triple.

a. 6, 8, 10

b. 5, 8, 9

c.