1. Pearson Unit 3 Topic 9: Similarity 9-4: Similarity in Right Triangles Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

2. TEKS Focus:
(8)(B) Identify and apply the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean to solve problems.
(1)(C) Select tools, including real objects, manipulatives paper and pencil, and technology as appropriate, and techniques, including mental math, estimations, and number sense as appropriate, to solve problems.
(1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
(7)(B) Apply the Angle-Angle criterion to verify similar triangles and apply the proportionality of the corresponding sides to solve problems.
(8)(A) Prove theorems about similar triangles, including the Triangle Proportionality theorem, and apply these theorems to solve problems.

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

5. Example: 1 By Theorem 9-3, ∆UVW ~ ∆UWZ ~ ∆WVZ.
Write a similarity statement comparing the three triangles.
Sketch the three right triangles with the angles of the triangles in corresponding positions. Z W
large
medium
small
By Theorem 9-3, ∆UVW ~ ∆UWZ ~ ∆WVZ.

6. Example: 2 By Theorem 9-3, ∆LJK ~ ∆JMK ~ ∆LMJ.
Write a similarity statement comparing the three triangles.
Sketch the three right triangles with the angles of the triangles in corresponding positions.
large
medium
small
By Theorem 9-3, ∆LJK ~ ∆JMK ~ ∆LMJ.

7. Consider the proportion
means
extremes
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.

8. 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 x2 = ab.

9. Example: 3
Find the geometric mean of each pair of numbers. If necessary, give the answer rounded to the nearest tenth.
4 and 25
Let x be the geometric mean. Write a proportion.
4 𝑥 = 𝑥 25
Def. of geometric mean
Cross multiply to solve the proportion.
x2 = (4)(25) = 100
x = 10
Find the positive square root.

10. Example: 4
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplified radical form and then also rounded to the nearest tenth.
5 and 30
Let x be the geometric mean.
5 𝑥 = 𝑥 30
Def. of geometric mean
x2 = (5)(30) = 150
Def. of geometric mean
Find the positive square root.
Simplify the square root.
x≈12.2
Rounded to the nearest tenth—only round if the directions tell you to round!

12. alt
seg 1 2
alt =
seg 1
OR:
seg 2

13. 2 8
Example: 5
Find BD

14. 3 8
Example: 6
Find BD

15. 10 5
Example: 7
Find DC and AC

16. 12
Example: 8
Find AD and AC = =

17. hyp
seg 1 =
leg
leg 2
leg 1
hyp
seg 2 =
leg
seg 1
seg 2
Hyp (AB)
hyp
adj. seg. =
leg
Generic:

18. Example: 9 5 4
Find BC

19. Example: 10 12 9
Find DA OR

20. Example: 11 7
Find BC

21. 12
Example: 12
Find AD

22. Example: 13 x = 12 x + 7 x(x+ 7) = 12(12) x2 + 7x = 144
Solve the following proportion.
x =
x + 7
x(x+ 7) = 12(12)
x2 + 7x = 144
x2 + 7x – 144 = 0
(x + 16) (x –9) = 0
x + 16 = x – 9 = 0
x = x = 9

23. Example: 14

24. Example: 15
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? x
The cliff is about , or 148 ft high