Trapezoid and similar triangles

Geometry Level 2

A isosceles trapezoid has bases A B = 15 AB=15 and C D = 20 CD=20 . Points E E and F F are on A D AD and B C BC respectively such that E F ∥ A B EF\parallel AB . If A E : E D = 2 : 3 AE:ED=2:3 , compute E F EF .


The answer is 17.

William Isoroku by William Isoroku , 25, USA

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1 solution

Chew-Seong Cheong
Apr 29, 2015

Let B H ∣ ∣ A D BH || AD and it intersects E F EF at G G .

Since A B ∣ ∣ E F ∣ ∣ C D AB || EF || CD ,

⇒ G F H C = A E A D = 2 5 ⇒ G F = 2 5 × H C = 2 ( 20 − 15 ) 5 = 2 \Rightarrow \dfrac {GF}{HC} = \dfrac {AE}{AD} = \dfrac {2}{5} \quad \Rightarrow GF = \dfrac {2}{5} \times HC = \dfrac{2 (20-15)}{5} = 2

⇒ E F = E G + G F = 15 + 2 = 17 \Rightarrow EF = EG + GF = 15+2 = \boxed{17}

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