Transcript Slide 1

Proportions and Similar
Triangles
Section 7.5
Objectives
• Use the Triangle Proportionality Theorem
and its converse.
Key Vocabulary
• Midsegment of a Triangle
Theorems
• 7.4 Triangle Proportionality Theorem
• 7.5 Converse of Triangle Proportionality
Theorem
• 7.6 Triangle Midsegment Theorem
Review Parallel Lines & Angle Pairs
• Corresponding ∠’s ≅
– Example: ∠11 ≅ ∠15
• Alternate Interior ∠’s ≅
– Example: ∠12 ≅ ∠15
• Alternate Exterior ∠’s ≅
– Example:∠10 ≅ ∠17
• Consecutive Interior ∠’s supplementary
– Example:∠12 & ∠14
• Vertical ∠’s ≅
– Example:∠14 ≅ ∠17
• Linear ∠’s supplementary
– Example:∠16 & ∠17
Proportional Parts Within
Triangles
• When a triangle contains a line that is
parallel to one of its sides, the two
triangles can be proved similar using the
AA shortcut.
• Example:
GH DF 
∠D≅∠EGH & ∠F≅∠EHG
∆DEF∼∆GEH
Corresponding ∠s
AA
• Since the triangles are similar, their sides
are proportional. Which leads us to the
Triangle Proportionality Theorem.
Theorem 7.4 Triangle
Proportionality Theorem
Triangle Proportionality
Theorem
If a line is parallel to one side
of a triangle and intersects
the other two sides, then it
divides the sides into
segments of proportional
lengths.
Example 1
Example 1
Substitute the known
measures.
Cross Products Property
Multiply.
Divide each side by 8.
Simplify.
Your Turn:
A. 2.29
B. 4.125
C. 12
D. 15.75
Example 2
Find the length of YZ.
44 36

35 ZY
 44  ZY    35 36 
44 ZY  1260
1260
ZY 
44
ZY  28.6
Example 3
Find Segment Lengths
Find the value of x.
SOLUTION
CD CE
=
DB EA
x
4
=
12
8
4 · 12 = 8 · x
48 = 8x
48 8x
=
8
8
6=x
Triangle Proportionality Theorem
Substitute 4 for CD, 8 for DB, x for CE,
and 12 for EA.
Cross product property
Multiply.
Divide each side by 8.
Simplify.
Example 4
Find Segment Lengths
Find the value of y.
SOLUTION
You know that PS = 20 and PT = y. By the Segment
Addition Postulate, TS = 20 – y.
PQ PT
=
QR
TS
y
3
=
9 20 – y
3(20 – y) = 9 · y
60 – 3y = 9y
Triangle Proportionality Theorem
Substitute 3 for PQ, 9 for QR, y for PT,
and (20 – y) for TS.
Cross product property
Distributive property
Example 4
Find Segment Lengths
60 – 3y + 3y = 9y + 3y
Add 3y to each side.
60 = 12y
Simplify.
60 12y
=
12 12
Divide each side by 12.
5=y
Simplify.
Your Turn
Find Segment Lengths and Determine Parallels
Find the value of the variable.
1.
ANSWER
8
ANSWER
10
2.
Proportional Parts Within
Triangles
• The converse of Theorem 7.4,
Triangle Proportionality Theorem, is
also true.
• Which is the next theorem.
Theorem 7.5 Converse of
Triangle Proportionality Theorem
Converse of the Triangle
Proportionality Theorem
If a line intersects two sides
of a triangle and separates
the sides into proportional
corresponding segments,
then the line is parallel to
the third side of the
triangle.
Example 5
In order to show that
we must show that
Example 5
Since
the sides are
proportional.
Answer:
Since the segments have
proportional lengths, GH || FE.
Your Turn:
A. yes
B. no
C. cannot be
determined
Example 6
Determine Parallels
Given the diagram, determine whether
MN is parallel to GH.
SOLUTION
Find and simplify the ratios of the two sides divided
by MN.
LM 56 8
=
=
MG 21 3
ANSWER
LN 48 3
=
=
NH 16 1
8
3 , MN is not parallel to GH.
Because ≠
3
1
Your Turn
Find Segment Lengths and Determine Parallels
Given the diagram, determine whether QR is parallel to ST.
Explain.
1.
ANSWER
2.
15 ≠ 17
no;
21 23
ANSWER
4
6
Yes; =
so QR || ST by the
8 12
Converse of the Triangle
Proportionality Theorem.
Triangle Midsegment
• A midsegment of a triangle is a segment that
joins the midpoints of two sides of the triangle.
• Every triangle has three midsegments, which
form the midsegment triangle.
• A special case of the triangle Proportionality
Theorem is the Triangle Midsegment
Theorem.
Theorem 7.6 Triangle Midsegment
Theorem
• A midsegment is a segment whose endpoints are the
midpoints of two sides of a Δ.
 Triangle Midsegment Theorem: A midsegment of a
triangle is parallel to one side of the triangle, and its
length is ½ the length of the side its parallel to.
If D and E are midpoints of
AB and AC respectively,
then DE || BC and
DE = ½ BC.
Example 7
Use the Midsegment Theorem
Find the length of QS.
SOLUTION
From the marks on the diagram, you know S is the
midpoint of RT, and Q is the midpoint of RP. Therefore,
QS is a midsegment of PRT. Use the Midsegment
Theorem to write the following equation.
QS =
1
1
PT = (10) = 5
2
2
ANSWER
The length of QS is 5.
Your Turn
Use the Midsegment Theorem
Find the value of the variable.
1.
2.
ANSWER
8
ANSWER
28
3. Use the Midsegment Theorem to find
the perimeter of ABC.
ANSWER
24
Example 8a
A. In the figure, DE and EF are midsegments of
ΔABC. Find AB.
Example 8a
1 AB
ED = __
2
1 AB
5 = __
2
Triangle Midsegment Theorem
10 = AB
Multiply each side by 2.
Answer: AB = 10
Substitution
Example 8b
B. In the figure, DE and EF are midsegments of
ΔABC. Find FE.
Example 8b
1 BC
FE = __
2
1 (18)
FE = __
2
Triangle Midsegment Theorem
FE = 9
Simplify.
Answer: FE = 9
Substitution
Example 8c
C. In the figure, DE and EF are midsegments of
ΔABC. Find mAFE.
Example 8c
By the Triangle Midsegment Theorem, AB || ED.
AFE  FED
Alternate Interior Angles Theorem
mAFE = mFED
Definition of congruence
AFE = 87
Substitution
Answer: AFE = 87°
Your Turn:
A. In the figure, DE and DF are midsegments of
ΔABC. Find BC.
A. 8
B. 15
C. 16
D. 30
Your Turn:
B. In the figure, DE and DF are midsegments of
ΔABC. Find DE.
A. 7.5
B. 8
C. 15
D. 16
Your Turn:
C. In the figure, DE and DF are midsegments of
ΔABC. Find mAFD.
A. 48
B. 58
C. 110
D. 122
Proportional Parts with Parallel
Lines
• Another special case of the Triangle
Proportionality Theorem involves three or
more parallel lines cut by two transversals.
• Notice that if transversals a and b are
extended, they form triangles with the
parallel lines.
Example 9
MAPS In the figure, Larch, Maple, and Nuthatch
Streets are all parallel. The figure shows the
distances in between city blocks. Find x.
Example 9
Notice that the streets form a triangle that
is cut by parallel lines. So you can use the
Triangle Proportionality Theorem.
Triangle Proportionality
Theorem
Cross Products Property
Multiply.
Divide each side by 13.
Answer: x = 32
Your Turn:
In the figure, Davis, Broad, and Main Streets are all
parallel. The figure shows the distances in between
city blocks. Find x.
A. 4
B. 5
C. 6
D. 7
Proportional Parts with Parallel
Lines
• If the scale factor of the proportional
segments is 1, then the parallel lines
separate the transversals into congruent
parts.
Congruent Parts of Parallel Lines
If three or more parallel lines cut off congruent
segments on one transversal, then they cut off
congruent segments on every transversal.
If AD BE CF and,
AB  BC , then DE  EF .
F
E
D
A
B
C
Example 10
ALGEBRA Find x and y.
To find x:
3x – 7 = x + 5
Given
2x – 7 = 5
Subtract x from each side.
2x = 12
x =6
Add 7 to each side.
Divide each side by 2.
Example 10
To find y:
The segments with lengths 9y – 2 and 6y + 4
are congruent since parallel lines that cut off
congruent segments on one transversal cut off
congruent segments on every transversal.
Example 10
9y – 2= 6y + 4
3y – 2 = 4
3y = 6
y =2
Answer: x = 6; y = 2
Definition of congruence
Subtract 6y from each side.
Add 2 to each side.
Divide each side by 3.
Your Turn:
Find a and b.
A.
2
; __
3
B. 1; 2
3
C. 11; __
2
D. 7; 3
Assignment
• Pg. 390 - 392 #1 – 29 odd
Joke Time
• How would you describe a frog with a
broken leg?
• Unhoppy
• What did the horse say when he got to the
bottom of his feed bag?
• That’s the last straw!