Circles
Circle
Allen Star Batch
Grade 11
Question:
In a triangle $ABC$, right angled at $A$, on the leg $AC$ as diameter, a semicircle is described. The chord joining $A$ with the point of intersection $D$ of the hypotenuse and the semicircle, then the length $AC$ equals to:
$\frac{AB \cdot AD}{\sqrt{AB^2 + AD^2}}$
$\frac{AB \cdot AD}{AB + AD}$
$\sqrt{AB \cdot AD}$
$\frac{AB \cdot AD}{\sqrt{AB^2 - AD^2}}$
Step-by-Step Solution
Key Concept: Apply the geometric mean altitude theorem and power of a point to relate the segments in the right triangle configuration.
Using the Pythagorean relation $(AC)^2 = (AD)^2 + (CD)^2$ and the power of point identity $(AB)^2 = (BD)(BC) = BD(BD + DC)$, we derive $DC = \frac{(AD)^2}{BC} = \frac{(AD)^2}{\sqrt{(AB)^2 - (AD)^2}}$. Finally, $AC = \frac{(AB)(AD)}{\sqrt{(AB)^2 - (AD)^2}}$.
Correct Answer: 4