Parallel lines and proportional parts
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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.

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Parallel Lines and Proportional Parts

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Parallel lines and proportional parts

Parallel Lines and Proportional Parts

Section 7.4


Proportional parts of triangles

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

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


Ab ed bd 8 dc 4 and ae 12

AB||ED, BD = 8, DC = 4, and AE = 12.

  • Find EC

  • by ?

  • EC = 6


Uy vx uv 3 uw 18 xw 16

UY|| VX, UV = 3, UW = 18, XW = 16.

  • Find YX.

  • YX = 3.2


Converse of side splitter

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


Determine if gh fe justify

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.


Parallel lines and proportional parts

  • Substitute

  • Simplify

  • Simplify


Parallel lines and proportional parts

  • Since the sides are proportional, then GH || FE.


Triangle midsegment theorem

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

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)


Parallel lines and proportional parts

  • 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


Parallel lines and proportional parts

  • Part 3 – Verify DE = ½ BC

  • To do this use the distance formula

  • BC = which simplifies to

  • DE =

  • DE = ½ BC


Corollaries of side splitter thm

Corollaries of side splitter thm.

  • 1. If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.


Parallel lines and proportional parts

  • If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.


Examples

Examples

  • 1. In the figure, Larch, Maple, and Nutthatch Streets are all parallel. The figure shows the distance between city blocks. Find x.


Parallel lines and proportional parts

  • 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


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