Transcript Slide 1

Transparency 6-5
5-Minute Check on Lesson 6-4
R
Refer to the figure
T
1. If QT = 5, TR = 4, and US = 6, find QU. 7.5
Q
S
U
2. If TQ = x + 1, TR = x – 1, QU = 10 and QS = 15, find x. 3
A
Refer to the figure
3. If AB = 5, ED = 8, BC = 11, and DC = x – 2,
find x so that BD // AE. 19.6
B
C
D
E
4. If AB = 4, BC = 7, ED = 5, and EC = 13.75, determine whether
BD // AE. Yes
5.
Find the
value of x + y in the figure?
Standardized Test Practice:
5y – 6
2y + 3
A
4
B
B
6
C
8
D
10
Click the mouse button or press the
Space Bar to display the answers.
3x – 2
2x + 1
Lesson 6-5
Parts of Similar Triangles
Objectives
• Recognize and use proportional relationships
of corresponding perimeters of similar
triangles
• Recognize and use proportional relationships
of corresponding angle bisectors, altitudes,
and medians of similar triangles
Vocabulary
• None New
Theorems
• If two triangles are similar then
– The perimeters are proportional to the measures of
corresponding sides
– The measures of the corresponding altitudes are proportional
to the measures of the corresponding sides
– The measures of the corresponding angle bisectors of the
triangles are proportional to the measures of the
corresponding sides
– The measures of the corresponding medians are proportional
to the measures of the corresponding sides
• Theorem 6.11: Angle Bisector Theorem: An angle
bisector in a triangle separates the opposite side into
segments that have the same ratio as the other two
sides
Special Segments of Similar Triangles
If ∆PMN ~ ∆PRQ, then
P
PM
PN
MN
special segment
----- = ----- = ----- = ------------------------AB
AC
BC
special segment
M
Q
N
ratios of corresponding special segments
= scaling factor
(just like the sides) in similar triangles
A
Example:
PM
1
----- = --AB
3
B
D
C
median PQ
1
----------------- = --median AD
3
Special segments are altitudes, medians,
angle and perpendicular bisectors
Similar Triangles -- Perimeters
If ∆PMN ~ ∆PRQ, then
P
M
R
Perimeter of ∆PMN
PM
PN
MN
------------------------- = ----- = ----- = ----Perimeter of ∆PRQ
PR
PQ
RQ
N
ratios of perimeters = scaling factor
(just like the sides)
Q
Angle Bisector Theorem - Ratios
P
If PN is an angle bisector of P, then the
ratio of the divided opposite side, RQ, is
the same as the ratio of the sides of P,
PR and PQ
PR
RN
----- = ----PQ
NQ
R
N
Q
Example 1a
If ∆ABC~∆XYZ, AC=32, AB=16, BC=165, and XY=24,
find the perimeter of ∆XYZ
C
Let x represent the perimeter of
The perimeter of
Example 1a cont
Proportional Perimeter
Theorem
Substitution
Cross products
Multiply.
Divide each side
by 16.
Answer: The perimeter of
Example 1b
If ∆PNO~∆XQR, PN=6, XQ=20, QR=202, and RX = 20,
find the perimeter of ∆PNO
R
Answer:
Example 2a
∆ ABC ~∆ MNO and 3BC = NO. Find the ratio of the
length of an altitude of ∆ABC to the length of an
altitude of ∆MNO
∆ ABC and ∆ MNO are similar with a ratio of 1:3.
(reverse of the numbers). According to Theorem 6.8,
if two triangles are similar, then the measures of the
corresponding altitudes are proportional to the
measures of the corresponding sides.
Answer: The ratio of the lengths of the altitudes is 1:3
or ⅓
Example 2b
∆EFG~∆MSY and 4EF = 5MS. Find the ratio of the
length of a median of ∆EFG to the length of a median
of ∆MSY.
Answer:
Example 3a
In the figure, ∆EFG~ ∆JKL, ED is an altitude of ∆EFG
and JI is an altitude of ∆JKL. Find x if EF=36, ED=18,
and JK=56.
Write a proportion.
K
Cross products
Divide each side by 36.
Answer: Thus, JI = 28.
Example 3b
In the figure, ∆ABD ~ ∆MNP and AC is an altitude of
∆ABD and MO is an altitude of ∆MNP. Find x if AC=5,
AB=7 and MO=12.5
N
Answer: 17.5
Example 4
The drawing below illustrates two poles supported by
wires with ∆ABC~∆GED , AFCF, and FGGC DC.
Find the height of the pole EC.
are medians of
since
and
If two triangles are similar, then the measures of the
corresponding medians are proportional to the measures
of the corresponding sides. This leads to the
proportion
Example 4 cont
measures 40 ft. Also, since
both measure 20 ft.
Therefore,
Write a proportion.
Cross products
Simplify.
Divide each side by 80.
Answer: The height of the pole is 15 feet.
Transparency 6-6
5-Minute Check on Lesson 6-5
Find the perimeter of the given triangle.
1. ∆UVW, if ∆UVW ~ ∆UVW, MN = 6, NP = 8, MP = 12, and UW = 15.6
33.8
2. ∆ABC, if ∆ABC ~ ∆DEF, BC = 4.5, EF = 9.9, and the perimeter of
∆DEF is 40.04. 18.2
Find x.
3.
2x
x=6
8
9
x–1
x = 7.375
9
4.
x
8.5
12
5.
Standardized Test Practice:
Find NO, if ∆MNO ~ ∆RSQ.
N
M
A
O
3.67
B
S
4.5 3
P
T
6.75
C
5.5
Q
7
R
D
8.25
Click the mouse button or press the
Space Bar to display the answers.
Find x and the perimeter of DEF,
if ∆DEF ~ ∆ABC
Find x
A
3x + 1
5x - 1
D
J
B
K
C
L
A
x
8
x-2
F
12
16
12
6
E
B
Find x if PT is an angle bisector
C
12
Find x, ED, and DB if ED = x – 3,
CA = 20, EC = 16, and DB = x + 5
Is CD a midsegment (connects two midpoints)?
P
x+2
8
E
C
R
S
6
T
x
Find y, if ∆ABC ~ ∆PNM
A
P
D
A
B
6
15
9
4
y
B
D
C
M
L
N
3x + 1
8
------- = ---5x – 1 12
Find x
A
3x + 1
5x - 1
36x + 12 = 40x – 8
20 = 4x
5=x
J
B
K
C
L
Find x and the perimeter of DEF,
x
6
if ∆DEF ~ ∆ABC
--- = ----
8
D
16
A
x
x-2
12
F
16
12
12
12x = 96
x=8
6
E
P = (x – 2) + x + 6
= 2x + 4
= 2(8) + 4
= 20
B
C
12
Find x, ED, and DB if ED = x – 3,
Find x if PT is an angle bisector
CA = 20, EC = 16, and DB = x + 5
8
6
Is CD a midsegment (connects two midpoints)?
------- = ---P
x+2
x
Since AC ≠ EC, then NO !
8x = 6x + 12
2x = 12 8
x=6
x+2
A
R
S
6
T
E
C
Find y, if ∆ABC ~ ∆PNM
D
P
x
A
6
4
------- = ---9
y
6y = 36
y=6
B
x–3
16
------- = ---x+5
20
6
15
9
4
y
B
D
C
M
L
N
20x - 60 = 16x + 80
4x = 140
x = 35
ED = 32
DB = 40
Summary & Homework
• Summary:
– Similar triangles have perimeters
proportional to the corresponding sides
– Corresponding angle bisectors, medians, and
altitudes of similar triangles have lengths in
the same ratio as corresponding sides
• Homework:
– Day 1: pg 319-20: 3-7, 11-15
– Day 2: pg 320-21: 17-19, 22-24,