Answer:
x = 9
Step-by-step explanation:
All of the right triangles in the figure are similar. That means ratios of corresponding sides are the same.
The two sides marked with numerical values have lengths 5 and 2√5 ≈ 4.47. The remainder of x, which is (x-5), will be shorter than either of the other two marked lengths. This means we can write the ratio of short side to long side of the two smaller triangles as ...
short side / long side = (2√5)/5 = (x-5)/(2√5)
Cross-multiplying gives ...
20 = 5(x -5)
4 = x -5 . . . . . divide by 5
9 = x . . . . . . . add 5
_____
Additional comment
In this geometry, as we have seen above, the square of the altitude ((2√5)²=20) is the product of the lengths of the segments of the hypotenuse (5×4=20). There are two other "geometric mean" relationships in a triangle like this, involving the legs of the largest triangle. They all arise from the similarity of the triangles.
