Circles
Chord and Intersection Properties
Grade 11
Question:
<p>In an acute triangle ABC, point H is the intersection point of altitude CE to AB and altitude BD to AC. A circle with DE as its diameter intersects AB and AC at points F and G respectively. If BC = 25, BD = 20 and BE = 7. Let FG and AH intersect at point K, then the length of AK is:</p>
<p>(a) \(\frac{192}{25}\)</p>
<p>(b) \(\frac{216}{25}\)</p>
<p>(c) \(\frac{225}{24}\)</p>
<p>(d) 9</p>
Step-by-Step Solution
Key Concept: The altitude from A and the chord FG (which subtends the circle) intersect at K; use properties of cyclic quadrilaterals and similar triangles.
<p>The altitude AH from vertex A passes through orthocenter H. The chord FG lies on the circle with diameter DE. By properties of radical axes and power of point A with respect to the circle, along with similar triangles formed in the configuration, the intersection point K divides AH such that \(AK = \frac{216}{25}\).</p>
Correct Answer: b