Proportional Parts

Advanced GeometrySimilarityLesson 4

If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates

these sides into segments of proportional lengths.

Triangle Proportionality Theorem

If ,BD AE

CB CD.

BA DE

endpoints are the midpoints of two sides

Midsegment

Triangle Midsegment TheoremA midsegment of a triangle is parallel to one side of the triangle,

and its length is one-half the length of that side.

BD AE

1

2BD AE

and

Example: Find x, BD, and AE.

Proportional SegmentsIf three or more parallel lines intersect two transversals,

then they cut off the transversals proportionally.

FJ GK HL������������������������������������������������������������������ �����

FG JK

GH KL

Example: Find x.

If two triangles are similar, then their perimeters areproportional to the measures of the corresponding sides.

Proportional Perimeters

EXAMPLE:If ∆DEF ∆GFH, find the perimeter of ∆DEF.∼

If two triangles are similar, then the measures of the corresponding altitudes, angle bisectors, and medians

are proportional to the measures of the corresponding sides.

Special Segments of Similar Triangles

EG

JL

EXAMPLE: In the figure, ∆EFD ~ ∆JKI. is a median of ∆EDF and

is a median of ∆JIK. Find JL if EF = 36, EG = 18, and JK = 56.

EXAMPLE: The drawing below illustrates two poles supported by wires. ∆ABC ~ ∆GED. CFAF FG GC DC.

EC. and

Find the height of pole

An angle bisector in a triangle separates the opposite side intosegments that have the same ratio as the other two sides.

Angle Bisectors

AD

AB

segments with

endpoint A

segments with

endpoint C

CD

CB

EXAMPLE:Find x if AB = 10, AD = 6, DC = x, and BC = x + 6.