5-Minute Check on Lesson 6-4 5-Minute Check on Lesson 6-4 Transparency 6-5 Click the mouse button or press the Click the mouse button or press the Space Bar to display the answers. Space Bar to display the answers. Refer to the figure 1. If QT = 5, TR = 4, and US = 6, find QU. 2. If TQ = x + 1, TR = x – 1, QU = 10 and QS = 15, find x. Refer to the figure 3. If AB = 5, ED = 8, BC = 11, and DC = x – 2, find x so that BD // AE. 4. If AB = 4, BC = 7, ED = 5, and EC = 13.75, determine whether BD // AE. 5. Find the value of x + y in the figure? Standardized Test Practice: A C B D 4 6 8 10 7.5 Yes 19.6 B 3 R T Q U S A B C D E 5y – 6 2y + 3 3x – 2 2x + 1
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5-Minute Check on Lesson 6-4 Transparency 6-5 Click the mouse button or press the Space Bar to display the answers. Refer to the figure 1.If QT = 5, TR.
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5-Minute Check on Lesson 6-45-Minute Check on Lesson 6-45-Minute Check on Lesson 6-45-Minute Check on Lesson 6-4 Transparency 6-5
Click the mouse button or press the Click the mouse button or press the Space Bar to display the answers.Space Bar to display the answers.
Refer to the figure
1. If QT = 5, TR = 4, and US = 6, find QU.
2. If TQ = x + 1, TR = x – 1, QU = 10 and QS = 15, find x.
Refer to the figure
3. If AB = 5, ED = 8, BC = 11, and DC = x – 2, find x so that BD // AE.
4. If AB = 4, BC = 7, ED = 5, and EC = 13.75, determine whetherBD // AE.
5. Find the value of x + y in the figure?Standardized Test Practice:
A CB D4 6 8 10
7.5
Yes
19.6
B
3
R
T
Q U S
A B C
D
E
5y – 6
2y + 3
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 TrianglesIf ∆PMN ~ ∆PRQ, then
PM PN MN special segment----- = ----- = ----- = -------------------------AB AC BC special segment
ratios of corresponding special segments = scaling factor(just like the sides) in similar triangles
Example:
PM 1 median PQ 1----- = --- ----------------- = ---AB 3 median AD 3
Special segments are altitudes, medians, angle and perpendicular bisectors
P
M
A
N
B CD
Q
Similar Triangles -- Perimeters
P
QR
M N
If ∆PMN ~ ∆PRQ, then
Perimeter of ∆PMN PM PN MN------------------------- = ----- = ----- = -----Perimeter of ∆PRQ PR PQ RQ
ratios of perimeters = scaling factor(just like the sides)
Angle Bisector Theorem - Ratios
P
QR N
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
If ∆ABC~∆XYZ, AC=32, AB=16, BC=165, and XY=24, find the perimeter of ∆XYZ
Let x represent the perimeter of The perimeter of
C
Example 1a
Proportional Perimeter Theorem
Substitution
Cross products
Multiply.
Divide each side by 16.
Answer: The perimeter of
Example 1a cont
If ∆PNO~∆XQR, PN=6, XQ=20, QR=202, and RX = 20, find the perimeter of ∆PNO
Answer:
R
Example 1b
∆ 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
∆ ABC ~∆ MNO and 3BC = NO. Find the ratio of the length of an altitude of ∆ABC to the length of an altitude of ∆MNO
Example 2a
Answer:
∆EFG~∆MSY and 4EF = 5MS. Find the ratio of the length of a median of ∆EFG to the length of a median of ∆MSY.
Example 2b
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.
K
Write a proportion.
Cross products
Divide each side by 36.
Answer: Thus, JI = 28.
Example 3a
Answer: 17.5
N
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
Example 3b
The drawing below illustrates two poles supported by wires with ∆ABC~∆GED , AFCF, and FGGC 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
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.
Example 4 cont
5-Minute Check on Lesson 6-55-Minute Check on Lesson 6-55-Minute Check on Lesson 6-55-Minute Check on Lesson 6-5 Transparency 6-6
Click the mouse button or press the Click the mouse button or press the Space Bar to display the answers.Space Bar to display the answers.
Find the perimeter of the given triangle.
1. ∆UVW, if ∆UVW ~ ∆UVW, MN = 6, NP = 8, MP = 12, and UW = 15.6
2. ∆ABC, if ∆ABC ~ ∆DEF, BC = 4.5, EF = 9.9, and the perimeter of ∆DEF is 40.04.
Find x.
3. 4.
5. Find NO, if ∆MNO ~ ∆RSQ.Standardized Test Practice:
A CB D3.67 6.75 7 8.25
33.8
x = 7.375 x = 6
D
18.2
2x9 8
x8.5
12
x – 1
9
5.5
RQ
S
T34.5
O P
N
M
B
C
A
K
L
J
3x + 1
5x - 1
8
12
A
B C
D
F
E
x
x - 2
6
12
12 16
Find x and the perimeter of DEF, if ∆DEF ~ ∆ABC
Find x
P
S
R
T
8
x6
x + 2
Find x if PT is an angle bisector
A
B
C
D
E
Find x, ED, and DB if ED = x – 3, CA = 20, EC = 16, and DB = x + 5
A
B
CD
6
4
P
M
N
y9
15
L
Find y, if ∆ABC ~ ∆PNM
Is CD a midsegment (connects two midpoints)?
B
C
A
K
L
J
3x + 1
5x - 1
8
12
A
B C
D
F
E
x
x - 2
6
12
12 16
Find x
P
S
R
T
8
x6
x + 2
Find x if PT is an angle bisector
A
B
C
D
E
Find x, ED, and DB if ED = x – 3, CA = 20, EC = 16, and DB = x + 5
A
B
CD
6
4
P
M
N
y9
15
L
Find y, if ∆ABC ~ ∆PNM
3x + 1 8------- = ----5x – 1 12
36x + 12 = 40x – 8 20 = 4x 5 = x
Find x and the perimeter of DEF, if ∆DEF ~ ∆ABC x 6
--- = ----16 12
12x = 96 x = 8
P = (x – 2) + x + 6 = 2x + 4 = 2(8) + 4 = 20
x – 3 16------- = ----x + 5 20
20x - 60 = 16x + 80 4x = 140 x = 35 ED = 32 DB = 40
8 6------- = ----x + 2 x
8x = 6x + 12 2x = 12 x = 6
6 4------- = ---- 9 y
6y = 36 y = 6
Is CD a midsegment (connects two midpoints)?Since AC ≠ EC, then NO !
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,