1. DO NOW:

2. 7.4: Similarity in Right Triangles
Similarity Shortcuts
7.4: Similarity in Right Triangles
Objectives:
I will be able to
find missing measures in similar right triangles
use shortcuts for prove that two triangles are similar

3. Geometric Mean
The geometric mean of two positive numbers a and b is the positive number x that satisfies This is just the square root of their product!

4. Example 1:

5. Now You Try:
a) What is the geometric mean of 8 and 12? b) What is the geometric mean of 4 and 6?

6. You can use geometry to measure distances that cannot be measured directly. Geometry provides a way to make these measurements indirectly through the use of the proportionality ratios that exist in similar triangles. This is the ancestor of trigonometry—the study of measurements using triangles.
Right triangles are exceptionally important in trigonometry because of the following: Two right triangles that contain congruent nonright angles are similar.

7. Thales
The Greek mathematician Thales was the first to measure the height of a pyramid by using geometry. He showed that the ratio of a pyramid to a staff was equal to the ratio of one shadow to another.

8. Example 2
If the shadow of the pyramid is 576 feet, the shadow of the staff is 6 feet, and the height of the staff is 5 feet, find the height of the pyramid.

9. Example 3
Explain why Thales’ method worked to find the height of the pyramid?

10. Example 4
If a person 5 feet tall casts a 6-foot shadow at the same time that a lamppost casts an 18-foot shadow, what is the height of the lamppost?

11. The shadow method only works outdoors on sunny days
The shadow method only works outdoors on sunny days. As an alternative, you can also use a mirror to estimate heights. The mirror method works both indoors and outdoors.

12. What if you decide to indirectly measure a height on a day when there are no shadows?
The mirror method is shown below. Place a mirror on a level spot at a convenient distance from the object. Back up from the mirror until you can see the top of the object in the center of the mirror.
The two triangles in the diagram are similar. To find the object’s height, you need to measure three distances and use similar triangles.

13. Example 5
Your eye is 168 centimeters from the ground and you are 114 centimeters from the mirror. The mirror is 570 centimeters from the flagpole. How tall is the flagpole?

14. www.youtube.com/watch?v=t9cxM_DQMXI \ Theorem 7.3

15. Example 6

16. Now You Try:
Write a similarity statement relating the three triangles in the diagram.
a) b)

17. http://www.youtube.com/watch?v=t9cxM_DQMXI Corollaries to Theorem 7.3

18. Corollary 1: Geometric Mean Altitude Theorem
Geometric Mean (Altitude) Theorem In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments.

19. Geometric Mean Theorem I
Heartbeat
In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments. x x a b

20. Corollary 2: Geometric Mean Leg Theorem
The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

21. Geometric Mean Leg Theorem
The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.
Boomerang a a x c

22. Geometric Mean Theorem II
Boomerang
Geometric Mean (Leg) Theorem The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. b b y c

23. Example 7

24. Now You Try:

25. Now You Try
e) A classmate writes the proportion to find b. Explain why the proportion is incorrect and provide the right answer.

26. Example 8
Solve for x and y. a)

27. Now You Try:
Solve for x and y. b)

28. Example 9
A quilter sews three right triangles together to make the rectangular quilt block at the right. What is the area of the rectangle? a) How can you find the dimensions of the rectangle? b) What is the formula for the area of a rectangle?

29. Now You Try:
A carpenter is framing a roof for a shed. What is the length of the longer slope of the roof?

30. Assignment
7.4 Homework Worksheet
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