The perimeter of isosceles trapezoid WXYZ is 62. If `bar(ZY)=7` and `bar(WX)=25`, what is the length of diagonal WY ?

Options

A. 15
B. 17
C. 20
D. 25

Answer

C

Explanation

As youread the stem, you might wonder what an isosceles trapezoid is. If you've never heard the term before, you still might have been able to extrapolate its meaning from what you know of isosceles triangles. Isosceles means ''having two equal sides''. When applied to a trapezoid, the term isosceles tells you that the two nonparallel sides - the legs - are equal. In this case, that's `bar(WZ)` and `bar(XY)`. If the total perimeter is 62, and the two marked sides add up to `25+7=32`, then the two unmarked sides split the difference of `62-32=30`. In other words, `bar(WZ)=bar(XY)=15` .

There aren't any special triangles yet. As so often happens, though, you can get some by constructing altitudes. Drop perpendiculars to points Z and Y, and you make two right triangles. The length 25 of side `bar(WX)` then gets split into 9, 7, and 9.

Now you can see that those right triangles are 3-4-5s (times 3) and that the height of the trapezoid is 12 because it is a leg of a `9-x-15` right triangle. Now look at the right triangle of which `bar(WY)` is the hypotenuse.

One leg is `7+9=16`, and the other leg is 12. The resulting triangle of which `bar(WY)` is the hypotenuse is another 3-4-5 triangle (times 4). `bar(WY)=20`, and the answer is (C ).