7.4 SIMILARITY IN RIGHT TRIANGLES

Done by Rana Karout

OBJECTIVES Find and use relationships in similar right triangles

HYPOTENUSEALTITUDE

Hypotenuse: In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle.

Altitude: The distance between a vertex of a triangle and the opposite side.

ACTIVITY: INVESTIGATING SIMILAR RIGHT TRIANGLES. DO IN PAIRS OR THREES.1. Cut an index card along one of its diagonals. 2. On one of the right triangles, draw an altitude

from the right angle to the hypotenuse. Cut along the altitude to form two right triangles.

3. You should now have three right triangles. Compare the triangles. What special property do they share? Explain.

4. Tape your group’s triangles to a piece of paper and write the conclusions.

THEOREM 7.3

A B

C

D

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.∆CBD ~ ∆ABC, ∆ACD ~ ∆ABC, ∆CBD ~ ∆ACD

GEOMETRIC MEAN

If BD=3cm and AD=5cm Find the length of CA, CB and CDDo you have enough information?Using the geometric mean formulas you can solve itWhat is the geometric mean?How can we know it and understand it and use it Keep this question in your mind….we will solve it , but later….

A B

C

D

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!

b

x

x

a =

abx =2 So

abx = And

EXAMPLE 1

Find the geometric mean of 12 and 27.

WRITE THIS DOWN!

A

C

BD

C D

B

A D

C

A C

B

BDCD

= CDAD

Shorter leg of ∆CBD.

Shorter leg of ∆ACD

Longer leg of ∆CBD.

Longer leg of ∆ACD.

GEOMETRIC MEAN THEOREMS

A

C

BDBDCD

= CDAD

ABCB

= CBDB

ABAC

= ACAD

A C

BA D

C

C D

B

USING THE GEOMETRIC MEAN

A B

C

D

A B

C

D

A B

C

D

DO YOU REMEMBER THIS!!!

If BD=3cm and AD=5cm Find the length of CA, CB and CD

A B

C

D

EXAMPLE

Find the value of x.

x

2712

Did you get x = 18?

EXAMPLE The altitude to the hypotenuse divides the hypotenuse into two segments.

What is the relationship between the altitude and these two segments?

x

2712

altitudealtitude

hypotenusehypotenuse

Segment 1 Segment 2