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Parallel Lines and Proportional Parts. Chapter 7-4. Use proportional parts of triangles. Divide a segment into parts. midsegment. - PowerPoint PPT Presentation
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Parallel Lines and Proportional Parts
Chapter 7-4
• midsegment
• Use proportional parts of triangles.• Divide a segment into parts.
Standard 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. (Key)
Triangle Proportionality Theorem• If a line parallel to one side of
a triangle intersects the other two sides, then it divides the two sides proportionally.
• The converse is true also.
B
A
E
DC
then DE // ifBEAB
CDACCB
DE //then , if CBBEAB
CDAC
Example #1?DE // IsCB
DE //then , If CBBEAB
CDAC
B
A
E
DC24
26
9.75
9
75.9
26924
9(26) 4(9.75)2
234342
DE // Yes CB
Find the Length of a Side
Find the Length of a Side
Substitute the known measures.
Cross products
Multiply.
Divide each side by 8.
Simplify.
A. 2.29
B. 4.125
C. 12
D. 15.75
Determine Parallel Lines
In order to show that we must show that
Determine Parallel Lines
Since the sides have
proportional length.
1. A2. B3. C
A. yes
B. no
C. cannot be determined
Midsegment Theorem• The midsegment connecting the midpoints
of two sides of the triangle is parallel to the third side and is half as long.
C
E
B
D
A
DE // AB
and
DE = AB21
Midsegment of a Triangle
Midsegment of a Triangle
Answer: D (0, 3), E (1, –1)
Use the Midpoint Formula to find the midpoints of
Midsegment of a Triangle
Midsegment of a Triangle
slope of
If the slopes of
slope of
Midsegment of a Triangle
Midsegment of a Triangle
First, use the Distance Formula to find BC and DE.
Midsegment of a Triangle
A. W (0, 1), Z (1, –3)
B. W (0, 2), Z (2, –3)
C. W (0, 3), Z (2, –3)
D. W (0, 2), Z (1, –3)
1. A2. B
A. yes
B. no
1. A2. BA. yes
B. no
Parallel Proportionality Theorem• If 3 // lines intersect two
transversals, then they divide the transversals proportionally.
then EF // CDAB// ifDFBD
CEAC
B
A
FD
C E
Example #2
P 9
UTS
QR
15
11
Find ST
SP // TQ // UR
Corresponding Angle Thm.
11915 x
Parallel Proportionality Theorem
355
91651659
x
x
Example #4J
K
M N
L7.5
9
13.5 x
y
37.5
Solve for x and y
What is JL? 37.5 – x
Solving for x
xx
5.37
5.139
)5.37(5.139 xx xx 5.1325.5069
25.5065.22 x5.22x
Example #4J
K
M N
L7.5
9
13.5 x
y
37.5
Solve for x and ySolving for yJKL~JMN
AA~Theorem
y5.22
5.79
75.1689 y75.18y
MAPS In the figure, Larch, Maple, and Nuthatch Streets are all parallel. The figure shows the distances in city blocks that the streets are apart. Find x.
Proportional Segments
Notice that the streets form a triangle that is cut by parallel lines. So you can use the Triangle Proportionality Theorem.
Answer: 32
Proportional Segments
Triangle Proportionality TheoremCross products
Multiply.
Divide each side by 13.
A. 4
B. 5
C. 6
D. 7
In the figure, Davis, Broad, and Main Streets are all parallel. The figure shows the distances in city blocks that the streets are apart. Find x.
Find x and y.
To find x:
Congruent Segments
GivenSubtract 2x from each side.Add 4 to each side.
To find y:
Congruent Segments
The segments with lengths are congruent
since parallel lines that cut off congruent segments on
one transversal cut off congruent segments on every
transversal.
Answer: x = 6; y = 3
Congruent Segments
Equal lengths
Multiply each side by 3 to eliminate the denominator.
Subtract 8y from each side.
Divide each side by 7.
Find a.
A.
B. 1
C. 11
D. 7
A. 0.5
B. 1.5
C. –6
D. 1
Find b.
HomeworkHomeworkChapter 7-4Chapter 7-4
•Pg 410Pg 41013-21, 26 – 13-21, 26 –
27, 32 – 36, 6127, 32 – 36, 61
