Test 7 Section 4 #36

Two important concepts in this question, closely related: trigonometry and Pythagorean triples.

First, the trig. The fact that means that the short legs of both triangles in the question are the lengths of the longer legs.

Now the Pythagorean triples: 3-4-5 is the most important Pythagorean triple to know! It’s called a Pythagorean triple because it’s a case where three integers work in the Pythagorean theorem: . If you know the legs of a right triangle are in a 3-4 ratio, then you know that you’ve got a 3-4-5 triangle (or one of its bigger cousins)!

The question tells you that *BC* = 15. That’s the hypotenuse of the longer side, so your larger triangle is a 9-12-15 (AKA a 3-4-5 times 3). So *AC* = 9 and *AB* = 12.

If *DA* = 4, that means *DB* = 12 – 4 = 8. This means the smaller of the triangles you’re dealing with is a 6-8-10 (AKA a 3-4-5 times 2). You’re asked for the value of *DE*, which must therefore be 6.