ACT question: In the figure below, ABC DFE, BAC FDE, D and F are on AB, AD FB, and distances in centimeters are as shown. What is the length of AD, in centimeters?
2 is the correct answer. DEF ~ ACB by AA (angle-angle similarity). Then, since AD = FB and corresponding sides of similar triangles are proportional, = = = = . Since you are shown that DF = 6, = ; 6(20) = 12(6 + 2AD); 120 = 72 + 24AD; 48 = 24AD. Therefore, AD = 2.
I assume that you understand how the triangles are similar;
`Delta DEF` ~ `Delta ACB`
Then since the triangles are similar, their corresponding sides are in proportion. (In other words, one of the triangles is a dilation of the other -- you multiply each of the sides by the same dilation constant. Then the ratio of corresponding sides is the same and is called the scale factor)
Since you didn't provide the numbers or figures, I also assume that you can tell from the diagram that either EF=12 and CB=20 or DE=12 and AC=20.
Then `Delta DEF` ~`Delta ACB ==> (DE)/(AC)=(EF)/(CB)=(DF)/(AB)`
We are given that DF=6 so AB=6+2AD. Substituting we get
`6/(6+2AD)=12/20` and the rest follows.