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

AB||ED, BD = 8, DC = 4, and AE = 12. Find EC by ? EC = 6

UY|| VX, UV = 3, UW = 18, XW = 16. Find YX. YX = 3.2

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 In triangle DEF, DH = 18, and HE = 36, and DG = ½ GF. To show GH || FE, Show Let GF = x, then DG = ½ x.

Substitute Simplify

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)

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

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