Begin by noting that the triangle on the left is a 5–12–13 Pythagorean
triple, so the bottom side is 5. Subtract 12.2 – 5 = 7.2 to get the bottom side of
the triangle on the right.
Next, the two unmarked angles that “touch” at the middle must sum to 90°,
because they form a straight line together with the right angle of 90° between
them, and all three angles must sum to 180°. Mark the angle on the left x. The
angle on the right must then be 90 – x.
Now the other angles that are still unmarked can be labeled in terms of x.
Using the rule that the angles in a triangle sum to 180°, the angle between 12
and 13 must be 90 – x, while the last angle on the right must be x,
Since each triangle has angles of 90, x, and 90 – x, the triangles are similar.
This observation is the key to the problem. Now you can make a proportion,
carefully tracking which side corresponds to which. The 7.2 corresponds to
12, since each side is across from angle x. Likewise, k corresponds to 13,
since each side is the hypotenuse. Write the equation and solve for k:
7.2/12=k/13
k=13*7.2/12
k=7.8