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9-5
Proportions in Triangles
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Exit Ticket
9-5
Proportions in Triangles
Warm Up #9
For each triangle, find the value of x.
1.
x=6
3.
x = 10
2.
x = 20
9-5
Proportions in Triangles
Find the value of x.
10
10 + 𝑥
𝑥 + 14
=
10
𝑥+2
10 + 𝑥 𝑥 + 2 = 10 𝑥 + 14
x+2
12𝑥 + 20 + 𝑥 2 = 10𝑥 + 140
𝑥 2 + 2𝑥 − 120 = 0
𝑥 + 12 𝑥 − 10 = 0
𝑥 = −12
𝑥 = 10
9-5
Proportions in Triangles
9-5
Proportions in Triangles
Connect to Mathematical Ideas (1)(F)
By the end of today’s lesson,
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.
9-5
Proportions in Triangles
Prior Knowledge:
Find the value of x.
12
9
=
𝑥 + 13
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
Example 1: Find the value of x.
𝑃𝐾
𝑁𝐿
∆ Proportionality Theorem.
=
𝐾𝑀
𝐿𝑀
𝑥+1
𝑥
=
12
9
Substitution
9 𝑥 + 1 = 12𝑥 Cross Products Property
9𝑥 + 9 = 12𝑥 Distributive Property
9 = 3𝑥
Subtract 9x from each side
𝑥=3
Divide 3 from each side
9-5
Proportions in Triangles
9-5
Proportions in Triangles
Example 2: Find the value of x.
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
Example 3: Identifying Similar Triangles.
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
𝑎
𝑥
= . In this case, the
Consider the proportion
𝑥
𝑏
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
𝑥=
2
𝑎𝑏, or 𝑥 = 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.
5 and 30
Let x be the geometric mean.
x2 = (5)(30) = 150
𝑥 =
150
𝑥 =5 6
Def. of geometric mean
Find the positive square root.
9-5
Proportions in Triangles
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.
𝒉𝟐 = 𝒚𝒙
𝒂𝟐 = 𝒄𝒙
𝒃𝟐 = 𝒄𝒚
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)
x=4
6 is the geometric mean of 9 and x.
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.
x = 7.5
c.
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?.
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.
When parallel lines intersect two or more segments, what is
the relationship between the segments formed?
The segments formed between the parallel lines are proportional.
9-5
Proportions in Triangles
Exit Ticket: Apply Mathematics (1)(A)
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?
Let x be the height of cliff above eye level.
(28)2 = 5.5x
28 is the geometric mean of 5.5 and x.
x  142.5
Divide both sides by 5.5.
The cliff is about 142.5 + 5.5, or 148 ft high.