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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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




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.




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)


  • 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



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.



Examples
Examples on one transversal, then they cut off congruent segments on every transversal.

  • 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. on one transversal, then they cut off congruent segments on every transversal.

  • 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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