Parallel Lines and Proportional Parts
-
Upload
camden-davenport -
Category
Documents
-
view
49 -
download
1
description
Transcript of Parallel Lines and Proportional Parts
Parallel Lines and Proportional Parts
Section 7.4
Proportional parts of triangles
• Non parallel transversals that intersect parallel lines can be extended to form similar triangles.
• So, the sides of the triangles are proportional.
Side Splitter Theorem or Triangle Proportionality Theorem
• If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally.
• If BE || CD, then
•
A
C D
B EAB AE
BC ED
AB||ED, BD = 8, DC = 4, and AE = 12.
• Find EC
• by ?
• EC = 6
C
B A
D E
CD CE
DB EA
4
8 12
EC
UY|| VX, UV = 3, UW = 18, XW = 16.
• Find YX.
• YX = 3.2
W
U
YX
VUV YX
WV XW
3
15 16
YX
Converse of Side Splitter
• If a line intersects the other two sides and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side.
• If
• then BE || CD
A
C D
B E
AB AE
BC ED
Determine if GH || FE. Justify
• In triangle DEF, DH = 18, and HE = 36, and DG = ½ GF.
• To show GH || FE,
• Show
• Let GF = x, then DG = ½ x.
DG DH
GF HE
• Substitute
• Simplify
• Simplify
118236
x
x
1182
1 36
1 1
2 2
• Since the sides are proportional, then GH || FE.
Triangle Midsegment Theorem
• A midsegment of a triangle is parallel to one side of the triangle, and its length is one-half the length of that side.
• If D and E are mid-
• Points of AB and AC,
• Then DE || BC and
• DE = ½ BC
Example
• Triangle ABC has vertices A(-2,2), B(2, 4) and C(4,-4). DE is the midsegment of triangle ABC.
• Find the coordinates of D and E.
• D midpt of AB• D(0,3)•
2 2 2 4,
2 2
• E midpt of AC
• E(1, -1)
• Part 2 - Verify BC || DE
• Do this by finding slopes
• Slope of BC = -4 and slope of DE = -4
• BC || DE
• Part 3 – Verify DE = ½ BC
• To do this use the distance formula
• BC = which simplifies to
• DE =
• DE = ½ BC
68 2 17
17
Corollaries of side splitter thm.
• 1. If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.
• If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.
Examples
• 1. In the figure, Larch, Maple, and Nutthatch Streets are all parallel. The figure shows the distance between city blocks. Find x.
• Find x and y.
• Given: AB = BC
• 3x + 4 = 6 – 2x
• X = 2
• Use the 2nd corollary to say DE = EF
• 3y = 5/3 y + 1
• Y = ¾