9-5 Proportions in Triangles

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9-5 Proportions in Triangles

Warm Up #9

For each triangle, find the value of x.

1. 2.

3.

x = 6 x = 20

x = 10

9-5 Proportions in Triangles

Find the value of x.

10

x + 2

10 + 𝑥

10=

𝑥 + 14

𝑥 + 2

10 + 𝑥 𝑥 + 2 = 10 𝑥 + 14

12𝑥 + 20 + 𝑥2 = 10𝑥 + 140

𝑥2 + 2𝑥 − 120 = 0

𝑥 + 12 𝑥 − 10 = 0

𝑥 = −12 𝑥 = 10

9-5 Proportions in Triangles

9-5 Proportions in Triangles

SWBAT

Use the Triangle Proportionality Theorem and the

Triangle-Angle-Bisector Theorem.

Use geometric mean to find segment lengths in

right triangles.

Apply similarity relationships in right triangles to

solve problems.

By the end of today’s lesson,

Connect to Mathematical Ideas (1)(F)

9-5 Proportions in Triangles

Prior Knowledge:

Find the value of x.

12

𝑥 + 13=

9

𝑥 + 9

12 𝑥 + 9 = 9 𝑥 + 13

12𝑥 + 108 = 9𝑥 + 117

3𝑥 = 9

𝑥 = 3

9-5 Proportions in Triangles

9-5 Proportions in Triangles

Find the value of x.

𝑥 + 1

12=

𝑥

9

9 𝑥 + 1 = 12𝑥

9𝑥 + 9 = 12𝑥

9 = 3𝑥

𝑥 = 3

Example 1:

∆ Proportionality Theorem.

Substitution

Cross Products Property

Distributive Property

𝑃𝐾

𝐾𝑀=

𝑁𝐿

𝐿𝑀

Subtract 9x from each side

Divide 3 from each side

9-5 Proportions in Triangles

9-5 Proportions in Triangles

Find the value of x.Example 2:

9-5 Proportions in Triangles

In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two right triangles.

9-5 Proportions in Triangles

9-5 Proportions in Triangles

Identifying Similar Triangles.Example 3:

What similarity statement can you write relating the three triangles in the diagram?

𝒀𝑾 is the altitude to the hypotenuse of right ∆XYZ, so you can use Theorem 9-3. There are three similar triangles.

9-5 Proportions in Triangles

Consider the proportion . In this case, the

means of the proportion are the same number, and

that number is the geometric mean of the extremes.

The geometric mean of two positive numbers is the

positive square root of their product. So the

geometric mean of a and b is the positive number x

such that , or .

𝑎

𝑥=

𝑥

𝑏

𝑥 = 𝑎𝑏 𝑥2 = 𝑎𝑏

9-5 Proportions in Triangles

9-5 Proportions in Triangles

Example 4: Finding Geometric Means

Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.

4 and 25

Let x be the geometric mean.

x2 = (4)(25) = 100 Def. of geometric mean

x = 10 Find the positive square root.

9-5 Proportions in Triangles

Example 5: Finding Geometric Means

Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.

Let x be the geometric mean.

5 and 30

x2 = (5)(30) = 150 Def. of geometric mean

Find the positive square root.𝑥 = 150

𝑥 = 5 6

9-5 Proportions in Triangles

𝒉𝟐 = 𝒚𝒙 𝒂𝟐 = 𝒄𝒙 𝒃𝟐 = 𝒄𝒚

You can use Theorem 9-3 to write proportions comparing the side lengths of the triangles formed by the altitude to the hypotenuse of a right triangle. All the relationships in red involve geometric means.

9-5 Proportions in Triangles

9-5 Proportions in Triangles

9-5 Proportions in Triangles

Example 6: Finding Side Lengths in Right Triangles

Find x, y, and z.

62 = (9)(x) 6 is the geometric mean of 9 and x.

x = 4 Divide both sides by 9.

y2 = (4)(13) = 52 y is the geometric mean of 4 and 13.

Find the positive square root.

z2 = (9)(13) = 117 z is the geometric mean of 9 and 13.

Find the positive square root.

9-5 Proportions in Triangles

Got It ? Solve With Your Partner

Problem 1 Finding the value of x.

a. b. c.

x = 7.5 x = 3.6 x = 35

9-5 Proportions in Triangles

Got It ? Solve With Your Partner

Problem 2 How far does the robot travel from A to D ?

9-5 Proportions in Triangles

Got It ? Solve With Your Partner

Problem 2 How far does the robot travel from A to D ?

9-5 Proportions in Triangles

Closure: Communicate Mathematical Ideas (1)(G)

How is the geometric mean used in right triangles?.

When parallel lines intersect two or more segments, what is the relationship between the segments formed?

The altitude of a right triangle to the hypotenuse is the geometric mean of the segments of the hypotenuse it creates. A leg of a right triangle is the geometric mean of the hypotenuse and the segments of the hypotenuse created by the altitude, adjacent to the leg.

The segments formed between the parallel lines are proportional.

9-5 Proportions in Triangles

A surveyor positions himself so that his line of sight to the top of a cliff and his line of sight to the bottom form a right angle as shown. What is the height of the cliff to the nearest foot?

Exit Ticket: Apply Mathematics (1)(A)

The cliff is about 142.5 + 5.5, or 148 ft high.

Let x be the height of cliff above eye level.

(28)2 = 5.5x 28 is the geometric mean of 5.5 and x.

Divide both sides by 5.5.x 142.5