Circles
Semicircle Properties
Grade 11
Question:
<p>In a right 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:</p>
<p>(a) \(\frac{AB \cdot AD}{AB^2 + AD^2}\)</p>
<p>(b) \(\frac{AB \cdot AD}{AB + AD}\)</p>
<p>(c) \(AB \cdot AD\)</p>
<p>(d) \(\frac{AB \cdot AD}{AB^2 - AD^2}\)</p>
Step-by-Step Solution
Key Concept: The angle inscribed in a semicircle is a right angle; use this property along with similar triangles formed in the configuration.
<p>This problem involves a right triangle with a semicircle inscribed on one leg. Using the property that the angle in a semicircle is a right angle and applying geometric relationships in the right triangle, the length AC can be derived from the relationship between AB and AD.</p>
Correct Answer: A