Download SimilarTrianglesProblem

Here is a problem that allows multiple areas of mathematics to be brought to the solution.
In the rectangle below, the three triangles, marked A, B, and C, are similar to each other.
The lengths of the sides of these triangles are proportional. Triangle C has legs of length
3 and 6. Find the lengths of the legs of triangles A and B.
Explain how you know the three triangles (using the legs you found) are similar.
A trigonometry solution
Using tan-1 and the Pythagorean Theorem, we can find the measures of the remaining
angles and the hypotenuse of triangle C. The angle measures are approximately 26.565°
from (tan-1 3/6) and 63.435° from tan-1 6/3, and the hypotenuse has measure about 6.708
from 6 2  32 . Using these angles in the similar triangles, triangle D is a right triangle
with acute angles of measures approximately 36.870° and 53.130° and one leg of
approximate length 6.708. Again using trigonometric ratios, we can find the lengths of
the three sides of that triangle. This gives us the length of one side in each of the other
two similar triangles. Since we know the angles in these right triangles, we can easily
find the lengths of the other sides.
A two-unknowns solution
Using the similar triangles allows us to identify the sides of x and 2x in triangle B and y
and 2y in triangle A. Using the opposite sides of the rectangle gives us two equations in x
and y:
3 + x = 2y
6 + y = 2x.
Solving two equations in these two unknowns gives x = 5 and y = 4.
A graphing solution
Place the rectangle on a coordinate grid as shown above, so two vertices of triangle D are
on the axes at (x, 0) and (0, 2x). Then the third vertex of triangle D is at (x + 6, 3).
Consider the line through (0, 2x) and (x + 6, 3), whose slope is -1/2.
3  2x
1
From the definition of slope,
, whose solution is x = 4.

( x  6)  0
2
From here the remaining lengths can be found.