7.1 The Pythagorean Theorem

Objectives/Assignment

Students will apply the Pythagorean Theorem

Use the Pythagorean Theorem to solve real-life problems such as determining how far a ladder will reach.

Mastery is 80% or better on 5 min checks and indy work.

Theorem 7.1 Pythagorean Theorem

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

ca

b

c2 = a2 + b2

Using the Pythagorean Theorem

A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c2 = a2 + b2 For example, the integers 3, 4 and 5 form a Pythagorean Triple because 52 = 32 + 42.

Ex. 1: Finding hypotenuse length

Find the length of the hypotenuse of the right triangle. Tell whether the sides lengths form a Pythagorean Triple.

x

12

5

Solution:

(hypotenuse)2 = (leg)2 + (leg)2

x2 = 52 + 122

x2 = 25 + 144 x2 = 169

x = 13

Because the side lengths 5, 12 and 13 are integers, they form a Pythagorean Triple. Many right triangles have side lengths that do not form a Pythagorean Triple as shown next slide.

Pythagorean TheoremSubstitute values.MultiplyAddFind the positive squareroot.

Note: There are no negative square roots until you get to Algebra II and introduced to “imaginary numbers.”

x

12

5

Ex. 2: Find leg lenthgs

Find the length of the leg of the right triangle.

x

14

7

Pythagorean Theorem

Substitute values.

Multiply

Subtract 49 from each side

Find the positive square root.

Use Product property

Simplify the radical.

(hypotenuse)2 = (leg)2 + (leg)2

142 = 72 + x2

196 = 49 + x2

147 = x2

√147 = x

√49 ∙ √3 = x

7√3 = x

Solution:

x

14

7

In example 2, the side length was written as a radical in the simplest form. In real-life problems, it is often more convenient to use a calculator to write a decimal approximation of the side length. For instance, in Example 2, x = 7 ∙√3 ≈ 12.1

Ex. 3: Find Triangle AreaFind the area of the triangle to the nearest tenth of a meter.

You are given that the base of the triangle is 10 meters, but you do not know the height.

h

10 m

7 m7 m

Because the triangle is isosceles, it can be divided into two congruent triangles with the given dimensions. Use the Pythagorean Theorem to find the value of h.

Solution:

Steps:

(hypotenuse)2 = (leg)2 + (leg)2

72 = 52 + h2

49 = 25 + h2

24 = h2

√24 = h

Reason:

Pythagorean Theorem

Substitute values.

Multiply

Subtract 25 both sides

Find the positive square root.

h

10 m

7 m7 m

Now find the area of the original triangle.

Area of a Triangle

Area = ½ b*h = ½ (10)(√24)

≈ 24.5 m2

The area of the triangle is about 24.5 m2

h

10 m

7 m7 m

Ex. 4: Indirect MeasurementSupport Beam: The skyscrapers

shown on page 535 are connected by a skywalk with support beams. You can use the Pythagorean Theorem to find the approximate length of each support beam.

Analyze

Each support beam forms the hypotenuse of a right triangle. The right triangles are congruent, so the support beams are the same length. Use the Pythagorean Theorem to show the length of each support beam (x).

Solution:

(hypotenuse)2 = (leg)2 + (leg)2

x2 = (23.26)2 + (47.57)2

x2 = √ (23.26)2 + (47.57)2

x ≈ 13

Pythagorean Theorem

Substitute values.

Multiply and find the positive square root.

Use a calculator to approximate.

What was the Objective?

Students will apply the Pythagorean Theorem

Use the Pythagorean Theorem to solve real-life problems such as determining how far a ladder will reach.

Mastery is 80% or better on 5 min checks and indy work.

HW

1-16 & 31-38 allPAGE 438