In the right triangle ABC above, BE=2 and AB=12. If the length of AC is 4 units more than 3 times the length of DE, what is the length
of DE ?
Approach
Typically, SAT triangle questions involve similarity. In this case, your intuition should tell you that the inner triangle is similar to the overall triangle. We can also prove this because we have two pairs of congruent angles.
m∠B≅m∠Breflexive propertym∠BED≅m∠BACright angles
It may be easier to figure out what's going if we decompose the initial figure into the individual triangles.
Since we know that △ABC∼△EBD, we can compare the side lengths to discover the relationship between corresponding sides.
Since BEBA=212=6, all of the corresponding sides of △ABC should have a length 6 times as large as △EBD.
Therefore,
AC=6⋅DE
The question also gives us the relationship:
AC=4+3DE
Substituting the value of AC from one equation into the other: