CK-12 Geometry-Concepts

http://www.ck12.org Chapter 7. Similarity

Investigation: Triangle Proportionality

Tools Needed: pencil, paper, ruler

1. Draw 4 ABC. Label the vertices.


2. DrawXYso thatXis onABandYis onBC.XandYcan beanywhereon these sides.


3. Is 4 X BY∼4ABC? Why or why not? MeasureAX,X B,BY,andY C. Then set up the ratiosAXX BandY CY B. Are
   they equal?


4. Draw a second triangle, 4 DEF. Label the vertices.


5. DrawXYso thatXis onDEandYis onEFANDXY||DF.


6. Is 4 X EY∼4DEF? Why or why not? MeasureDX,X E,EY,andY F. Then set up the ratiosDXX EandY EFY. Are
   they equal?

From this investigation, it is clear that if the line segments are parallel, thenXYdivides the sides proportionally.

Triangle Proportionality Theorem:If a line parallel to one side of a triangle intersects the other two sides, then it
divides those sides proportionally.

Triangle Proportionality Theorem Converse: If a line divides two sides of a triangle proportionally, then it is
parallel to the third side.

Proof of the Triangle Proportionality Theorem: