7.4 Similarity in Right Triangles

1. 7.4 SIMILARITY IN RIGHT
TRIANGLES
Done by Rana Karout

2. OBJECTIVES Find and use relationships in
similar right triangles

3. HYPOTENUSE
ALTITUDE
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.

4. 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
fromthe right angle to the hypotenuse. Cut along
the altitude to formtwo right triangles.
3. You should now have three right triangles.
Compare the triangles. What special property do
they share? Explain.
4. Tape yourgroup’s triangles to a piece of paper
and write the conclusions.

5. 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

6. GEOMETRIC
MEAN
If BD=3cm and AD=5cm
Find the length of CA, CB and CD
Do you have enough information?
Using the geometric mean formulas
you can solve it
What 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

7. 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

8. EXAMPLE 1
Find the geometric mean of 12 and 27.

9. WRITE THIS DOWN!
A
C
BD
C D
B
A D
C
A C
B
BD
CD
=
CD
AD
Shorter leg of ∆CBD.
Shorter leg of ∆ACD
Longer leg of ∆CBD.
Longer leg of ∆ACD.

10. GEOMETRIC MEAN
THEOREMS
A
C
BD
BD
CD
=
CD
AD
AB
CB
=
CB
DB
AB
AC
=
AC
AD
A C
B
A D
C
C D
B

11. USING THE GEOMETRIC MEAN
A B
C
D
A B
C
D
A B
C
D

12. DO YOU
REMEMBER
THIS!!!If BD=3cm and AD=5cm
Find the length of CA, CB and CD
A B
C
D

13. EXAMPLE
Find the value of x.
x
2712
Did you get x = 18?

14. 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