+51 In triangle XYZ, XY=20, YZ=15, and XZ=25. Y is reflected over line XZ to Y'. Find YY'.
+309 Answer: 24
Solution:
Note that XYZ is a right triangle where Y is 90 degrees. Y would be reflected over XZ at the point closest to XZ, which is the foot of the altitude from XZ to Y, which means that YY' would be double that altitude. You may be accustomed to using the altitude of a triangle to find its area, but it can also be derived from the area. Because XYZ is a right triangle, its area can be found by multiplying its legs and then dividing by two.
15 x 20 / 2 = 150
Because the area of the triangle is an altitude multiplied by the side its foot is on over two, the desired altitude would be multiplied by XZ (25) and then divided by two to get the area of the triangle. Set the desired altitude's length to a:
25a / 2 = 150
a = 12
YY' is double this length, YY' is equal to 24.
