Transcript Use Similar Right Triangles

Use Similar Right Triangles
Ch 7.3
Similar Right Triangle Theorem
• If the altitude is drawn to the hypotenuse of a
right triangle, then the two triangles formed
are similar to the original right triangle.
How do you names the 3 similar
triangles?
1. Draw the smallest triangle.
2. Draw the middle triangle.
3. Draw the largest triangle.
4. Match up the angles.
îSUT ~ îTUR ~ îSTR
Name the similar triangles, then find x.
îEHG ~ îGHF ~ îEGF
To find x make a ratio of the
hypotenuses and the a ratio of 2
proportional legs.
EG GH

EF GF
3 x
5
x

12

5 4
12
x
5
Name the similar triangles and find x.
îLKM ~ îMKJ ~ îLMJ
Hypotenuse
2nd Longest side
JL
LM 13 5


JM KM 12 x
13 x  60
60
x
13
Find x and y.
72
y
x
Hyp
nd
2 longest
2
2
2
AB CB
a b  c

21
AC CD
2
2
2
21  72  c
75 21

2
441 5182 c
72 y
2
75y  1512
5625 c
y  20.16
75  c
2
2
2
x  20.16  72
x  69.12
Find x.
Shortest
nd
2 longest
3 x

x 12
2
x  36
x  36
2
x6
Find x
x 4

5 x
2
x  20
x  20 
2
5 4 
5 2 2 
2 5
Theorem 7.6
• In a right triangle the altitude from the right
angle to the hypotenuse divides the
hypotenuse into 2 segments.
• The length of the altitude is the geometric
mean of the lengths of the 2 segments
CD DB

AD CD
CD  AD DB
y  2  8  16  4
8  x4
8  ( x  4)
2
64  4 x
16  x
2
Finding the length of the altitude
B
D
C
1.Set up a proportion to find
BD.
2. Find side AD.
3. Plug values into
A
CD  AD  DB
Finding the length of the altitude.
Big Triangle Hypotenuse Smallest Triangle Hypotenuse

Big Triangle SmallestSide Smallest Triangle SmallestSide
10.8
19.2
30 18

18 x
30x  (18)(18)
30x  324
x  10.8
30  10.8  19.2
Altitude 19.2 10.8  207.36  14.4
Finding the length of the altitude.
Big Triangle Hypotenuse Smallest Triangle Hypotenuse

Big Triangle SmallestSide Smallest Triangle SmallestSide
10.7
2.7
6 5 6

6
x
6 5 x  (6)( 6)
6 5 x  36
x  2 .7
6 5  2.7  10.7
Altitude 10.7  2.7  28.89  5.3
Finding the length of the altitude.
Big Triangle Hypotenuse Smallest Triangle Hypotenuse

Big Triangle SmallestSide Smallest Triangle SmallestSide
9.6
13.8
3 61 15

15
x
3 61x  (15)(15)
3 61x  225
x  9 .6
3 61  9.6  13.8
Altitude 13.8  9.6  132.48  11.5
Find the amplitude, if these are right
triangles. One of these is not a right
triangle
8.5
6.6
not right
Theorem 7.7
• In a right triangle, the altitude divides the
hypotenuse into 2 segments.
• The length of each leg of each right triangle is
the geometric mean of length of the
hypotenuse and a segment of the hypotenuse
CB  AB  DB
AC  AB  AD
Find x and y
y
12.75
x
CB  AB  DB
CB  17 4.25
CB  72.25  8.5
4.25
AC  AB  AD
AC  1712.75
AC  216.75  14.7
Find x
x  8 2
x  16
x4
Find x and y
y
x +2
8
2
AC  AB  AD
AC  10 8
AC  80  8.9
CD  AD  DB
x  2  8 2
x  2  16
x24
x2
Find a
Find b
Find x and y
30
16
z
|---------
x
34
y
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