Chapter 10.4, Problem 2PSA

To determine

To Find : $\overline{XY}$

Answer to Problem 2PSA

  $\overline{XY}\cong 12\text{ unit}$

Explanation of Solution

Given information : $\begin{array}{l}\text{Concentric circles with centre P and radii 8 and 10}\text{.}\\ \overline{\text{XY}}\text{ is tangent to inner circle}\text{.}\end{array}$

  $\begin{array}{l}PZ=8andPY=10.\\ As\overline{XY}\text{ is chord of the bigger circle}\text{.}\\ \Rightarrow \text{PZ}\perp XYandXZ=YZ\\ \text{Because perpendicular from centre bisector the chord}\\ \Rightarrow \Delta \text{PZ}Y=90°\\ In\Delta PZY,\\ P{Y}^2\cong P{Z}^2+Z{Y}^2\\ \Rightarrow P{Y}^2\cong 8^2+2Y^2\\ \Rightarrow {10}^2\cong 8^2+2Y^2\\ \Rightarrow {\overline{ZY}}^2\cong 100-64=36\\ \Rightarrow \overline{ZY}\cong 6\\ \Rightarrow \overline{XY}\cong 2\overline{ZY}\\ \Rightarrow \overline{XY}\cong 2\times 6=12\end{array}$