Notice that these two triangles share both a 90-degree angle AND the angle I've denoted with a purple dot. Since the 3 angles in each triangle must add to 180 degrees, we know that their 3rd angle must also be equal.
So, the two triangles are similar. From here, we can use ratios to find the value of x, or we can use something I called the "enlargement factor." Both approaches use the same idea, but I think the "enlargement factor" is easier to grasp.
First notice that the blue sides are corresponding sides, since they are both the hypotenuse of their respective triangles.
The hypotenuse of the larger triangle has length 12, and the hypotenuse of the smaller triangle has length 7.
So, the larger triangle is 12/7 larger than the small triangle. In other words, the enlargement factor is 12/7
Our goal is to find the value of x, which represents the length of the red side of the large triangle. We know that this side will be 12/7 longer than the red side of the small triangle (which has length 3)
So, x = (12/7)(3) = [spoiler]36/7[/spoiler]
Cheers,
Brent
