What is the number of centimeters in the length of \(EF\) if \(AB\parallel CD\parallel EF\)?

tertre
Mar 20, 2018

#2**+2 **

To avoid getting confused by the apparent right angles, I changed the picture to make BC more slanted looking.

Since alternate interior angles are congruent....

∠CAB = ∠ACD and

∠ABD = ∠BDC

So by AA similarity, △ABE ~ △CDE

So if ED = a, then EB = 1.5a

Since corresponding angles are congruent....

∠BDC = ∠BEF and

∠BCD = ∠BFE

So by AA similarity, △BCD ~ △BFE

FE / CD = BE / BD

FE / 100 = 1.5a / (1.5a + a)

FE = 100 * 1.5a / (1.5a + a)

FE = 60 cm

hectictar
Mar 20, 2018

#3**+2 **

Note that angle BAE = angle ECD

And angle ABE = angle EDC

So....by AA congruency, triangle ABE is similar to triangle CDE

And CD : AB = 2: 3

Then the height of CDE = CF is 2/3 height of ABE = FB

So.... CF : FB = 2 : 3

And angle BFE = angle BCD

And angle BEF = angle BDC

So,again, by AA congruency, triangle BDC is similar to triangle BEF

But there are 5 parts of BC....and BF is 3/5 of these

So......BF is 3/5 of BC

So....EF is 3/5 of DC = (3/5) (100) = 60 (cm)

CPhill
Mar 20, 2018