Right Triangle Similarity
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Transcript of Right Triangle Similarity
- 1. SIMILARITY IN RIGHT TRIANGLES
2. We start with ABC.
3. We draw altitude CD to the hypotenuse.
4. This divides the original triangle into two smaller right
triangle:
5. This divides the original triangle into two smaller right
triangle: DCA
6. This divides the original triangle into two smaller right
triangle: BDC
7. There are a three triangles in the figure below.
8. Big
Medium
Small
9. Big
Medium
Small
10. We orient the three triangles to see the them clearer.
Big
Small
Medium
11. We can see that the three triangles are similar to each
other.
~
~
Big
Small
Medium
12. SIMILARITY IN RIGHT TRIANGLES
13. Parts of a right triangle incorporated with the altitude
C
Leg adjacent to DB
Leg adjacent to AD
A
B
D
Segments of the hypotenuse AD and DB
14. Right Triangle Similarity Theorem
The altitude to the hypotenuse of a right triangle divides the
triangle into two triangles that are similar to the original
triangle and to each other.
C
A
B
D
ABC ~ ACD ~ CBD
15. Geometric Mean-Altitude Theorem 1
The length of the altitude to the hypotenuse is the geometric mean
of the lengths of the segments of the hypotenuse.
C
A
B
D
=
=
16. Geometric Mean-Altitude Theorem 2
The altitude to the hypotenuse to a right triangle intersects it to
that the length of each leg us the geometric mean of the length of
its adjacent segment of the hypotenuse and the length of the entire
hypotenuse
BACB=CBBD
ABCA=CAAD
=
=
17. Summary of theGeometric Mean Altitude Theorem
b
a
h
m
n
c
=+
=
=
=
18. Solve for the other missing lengths given only two
measurements.
a = 4, b = 6
a = 8, c = 10
a = 5, m = 7
a = 9, n = 6
a = 12, h = 9
b = 6, c = 15
b = 8, m = 9
b = 4, n = 3
b = 11, h = 8
c = 18, m = 12
c = 15, n = 8
c = 20, h = 6
m = 12, n = 8
m = 9, h = 12
n = 10, h = 12
19. Solve for the other missing lengths given only two
measurements.
a = 4, b = 6
a = 8, c = 10
a = 5, c = 8
a = 9, m = 6
a = 12, h = 9
b = 6, c = 15
b = 12, n = 9
b = 4, n = 3
b = 10, h = 6
b = 11, h = 8
c = 18, m = 12
c = 15, n = 8
c = 20, m = 6
m = 12, n = 8
m = 9, h = 12
n = 10, h = 12
a = 9, m = 6
c = 18, m = 12
20. Relating to the Real WorldRecreation
At the parking lot of a State Park, the 300-m path to the snack bar
and the 400-m path to the boat rental shop meet at a right angle.
Marla walks straight from the parking lot to the ocean.How far is
Marla from the snack bar?
WHICH IS STRONGER? TRIANGLE OR QUADRILATERAL?