Given Parallelogram ABCD in which diagonals meet at G.
A segment is constructed through G intersecting BC at E and AD at F.
mBE = 2.0 in.
mAF = 2.8 in.
Need to find measures of BC and AD.
In geometry, the first step is to draw a diagram according to given information, if one is not already provided. (see attachment).
Next, based on basic theorems, deduce as much information as possible, and take note.
Properties of a parallelogram include (but not limited to):
- opposite sides are parallel .................(1)
- opposite sides are congruent. ............(2)
- diagonals bisect each other.................(3)
Strategy:
prove congruence of triangles BEG and DFG
=> BE=FD
=> mAD=2.8+2.0=4.8 = mBC
Congruence of BEG and DFG
Consider transversal BD, we conclude that
angle EBG is congruent to angle FDG (alternate interior angles)...(A)
Consider diagonal BD,
mBG = mGD (property (3) above)..................................................(S)
Consider transversal EF, we conclude that
angle BEG is congruent to angle FGD (alternate interior angles)....(A)
=> triangle BEG is congruent to triangle DFG (ASA).
=> FD=BE=2.0 ............(corresponding sides of congruent triangles)
=> AD=2.8+2.0=4.8
=> BC=AD=4.8........................(property (2) above).
Answer: BC=AD=4.8 inches.
