Download Practice 7-5: PROPORTIONS IN TRIANGLES

PRACTICE 7-5 (DNG PAGE 387)
7-5 Guided Problem Solving (page 388)
Proportions in Triangles
1
19 January 2014
Geometry/CBautista
Practice 7-5: PROPORTIONS IN TRIANGLES
(DNG - page 387)
Use the figure at the right to complete each proportion.
BE
EH
JD
BC
JG
BE
JI
The Side-Splitter Theorem states that if a line is || to one side of a  and
intersects the other two sides, then it divides those sides proportionally.
COROLLARY: If three parallel lines intersect two transversals,
then the segments intercepted on the transversals are proportional.
The  -- Bisector Theorem states that if a ray bisects an  of a ,
then it divides the opposite side into two segments that are
proportional to the other two sides of the triangle.
Practice 7-5: PROPORTIONS IN TRIANGLES
Algebra
Find the values of the variables.
𝒙
𝟔
=
𝟖
𝟗
𝒙=𝟖
𝒙=
𝟏𝟔
𝟑
DNG - page 387
𝒙
𝟒
=
𝟓
𝟓
𝟔
𝟗
𝒙=𝟓
𝒙
𝟐
=
𝟐
𝟏
𝟒
𝟓
𝒙=𝟐
𝒙=4
𝑥=4
The  -- Bisector Theorem states that if a ray bisects an  of a ,
then it divides the opposite side into two segments that are
proportional to the other two sides of the triangle.
COROLLARY: If three parallel lines intersect two transversals,
then the segments intercepted on the transversals are proportional.
𝟐
𝟏
Practice 7-5: PROPORTIONS IN TRIANGLES
DNG - page 387
Algebra Find the values of the variables.
=
𝟓
𝟑
+
𝟒
𝟑
𝟐𝟓
𝟗
=
25
9
20
9
+
=5
=
=𝟑
Solve for x :
𝒙
𝟏𝟎
=
𝟏𝟎
𝟖
𝒙 = 𝟏𝟎
𝒙=
𝟐𝟓
𝟐
𝟏𝟎
𝟖
𝒙
=
𝟓
𝟑
𝟓
𝒙=
𝟑
𝒙=
Solve for y :
Pythagorean Theorem :
𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐
𝟑𝟐 + 𝒚𝟐 = 𝟓𝟐
𝟐𝟎
𝟗
𝟒
𝟑
𝟐𝟎
𝟗
𝟑
𝟒
𝟐𝟓
𝟗
The  -- Bisector Theorem states
that if a ray bisects an  of a , then
it divides the opposite side into two
segments that are proportional to the
other two sides of the triangle.
𝟗 + 𝒚𝟐 = 25
𝒚𝟐 = 16
𝒙
𝟑
=
𝟓
𝟒
𝒙=𝟓
𝒙=
𝒚𝟐 = 𝟏𝟔
𝒚=4
The Side-Splitter Theorem states
that if a line is || to one side of a  &
intersects the other two sides, then
it divides those sides proportionally.
𝟏𝟓
𝟒
𝟑
𝟒
Practice 7-5: PROPORTIONS IN TRIANGLES
DNG - page 385
Algebra Find the values of the variables.
|
¬
|
Since the  is isosceles,
the  bisector is also
a  bisector.
∴ 𝒙 = 𝟔; 𝒚 = 𝟔
𝒙
𝟐𝟏
=
𝟐𝟎
𝟑𝟔
𝟐𝟏
𝒙 = 𝟑𝟔
𝟐𝟎
𝟏𝟖𝟗
𝒙=
𝟓
𝒚
𝟐𝟐
=
𝟐𝟎
𝟑𝟔
𝟐𝟐
𝒚 = 𝟑𝟔
𝟐𝟎
𝟏𝟗𝟖
y=
𝟓
COROLLARY: If three parallel lines intersect two transversals,
then the segments intercepted on the transversals are proportional.
Practice 7-5: PROPORTIONS IN TRIANGLES
DNG - page 385
Algebra Solve for x.
𝒙
=
𝒙+𝟏
𝟔
𝟗
Write the proportion.
𝟗𝒙 = 𝟔 𝒙 + 𝟏
Cross-Product Property
𝟗𝒙 = 𝟔𝒙 + 𝟔
Distributive Property
𝟑𝒙 = 𝟔
Subtract 𝟔𝒙 from each side.
𝒙=𝟐
Divide by 3.
The Side-Splitter Theorem states that if a line is || to one side of a  and
intersects the other two sides, then it divides those sides proportionally.
Practice 7-5: PROPORTIONS IN TRIANGLES
DNG - page 387
Algebra Solve for x.
𝒙
=
𝒙+𝟒
𝒙−𝟏
Write the proportion.
𝒙+𝟐
𝒙 𝒙+𝟐 = 𝒙+𝟒 𝒙−𝟏
Cross-Product Property
𝒙𝟐 + 𝟐𝒙 = 𝒙𝟐 −𝒙 + 𝟒𝒙 − 𝟒
Distributive Property
𝒙𝟐 + 𝟐𝒙 =
Simplify.
𝒙𝟐 + 𝟑𝒙 − 𝟒
−𝒙𝟐 − 𝟑𝒙 = −𝒙𝟐 − 𝟑𝒙
−𝒙 = −𝟒
𝒙=4
The  -- Bisector Theorem states that if a ray bisects an  of a ,
then it divides the opposite side into two segments that are
proportional to the other two sides of the triangle.
Practice 7-5: PROPORTIONS IN TRIANGLES
Algebra Solve for x.
COROLLARY: If three parallel
lines intersect two transversals,
then the segments intercepted on
the transversals are proportional.
𝒙
𝒙+𝟓
=
𝟐𝒙 − 𝟖
𝒙+𝟖
Write the proportion.
𝒙 𝒙 + 𝟖 = 𝟐𝒙 − 𝟖 𝒙 + 𝟓
Cross-Product Property
𝒙𝟐 + 𝟖𝒙 = 𝟐𝒙𝟐 +𝟏𝟎𝒙 𝟖𝒙 − 𝟒𝟎 Distributive Property
𝒙𝟐 + 𝟖𝒙 = 𝟐𝒙𝟐 + 𝟐𝒙 − 𝟒𝟎
Simplify.
−𝒙𝟐 − 𝟖𝒙 = −𝒙𝟐 − 𝟖𝒙
𝟎 = 𝒙𝟐 − 𝟔𝒙 − 𝟒𝟎
𝟎 =
𝒙+𝟒=𝟎
𝒙 = −𝟒
*not a possible answer,
Length cannot be negative.
𝒙 +𝟒
𝒙 − 𝟏𝟎 Factor.
𝒙 − 𝟏𝟎 = 𝟎
𝒙 = 10
Equate each factor to 0
and solve for x.
ANSWER.
7-5 • Guided Problem Solving
(DNG - page 388)
Determine whether the segments 𝑪𝑫 and 𝑨𝑩 are ||.
You can use the Converse of the Side-Splitter Theorem.
Read and Understand
1. The Converse of the Side-Splitter Theorem states:
If a line divides two sides of a  proportionally,
parallel to the third side.
then it is _________
The Side-Splitter Theorem states
that if a line is || to one side of a 
and intersects the other two
sides, then it divides those sides
proportionally.
RECALL  Conditional: 𝒑 → 𝒒
Converse: 𝒒 → 𝒑
Plan and Solve
You need to show 𝑪𝑫 || 𝑨𝑩 .
To do this, show the corresponding parts divided by the line are proportional.
CE
𝑨𝑪
2. Write a proportion.
=
DB
𝑫𝑬
𝟏𝟎
15
=
90 =
6
𝟗
90
3. Substitute.
4. Use the Cross-Product Property.
5. Are the two sides of the equation the same? YES
6. Therefore, are 𝑪𝑫 and 𝑨𝑩 parallel? YES
Look Back and Check
7. If your answer to Step 5 were different, how would that change your answer to Step 6?
The sides will not be proportional.
Solve Another Problem
8. Based on your answer to Step 6, how are AEB and CED related? They are similar triangles.