Question

If U, V, equals, 72UV=72, V, T, equals, 82VT=82, and Y, W, equals, 49YW=49, find the length of start overline, X, Y, end overline
XY
. Round your answer to the nearest tenth if necessary. Figures are not necessarily drawn to scale.

Answer 1

Interpret X as the external point that creates two secants: one through U and T (with XU = 72, XT = 82) and the other through Y and W (with YW = 49). By the power-of-a-point (secant) theorem,

XU · XT = XY · XW = XY · (XY + YW).

So
72 · 82 = XY( XY + 49 ).

Compute 72·82 = 5904, so
XY^2 + 49XY − 5904 = 0.

Solve for the positive root:
XY = [−49 + sqrt(49^2 + 4·5904)] / 2
= [−49 + sqrt(26017)] / 2
≈ (−49 + 161.2457)/2
≈ 56.12285.

Rounded to the nearest tenth: XY ≈ 56.1.