Question

On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, is

Answer 1

Correct Answer: 24

Refer to the diagram:

Observe that triangles \( \triangle BPC \) and \( \triangle BQC \) are inscribed in a semicircle. Hence, by the property of semicircles:

\( \angle BPC = \angle BQC = 90^\circ \)

Therefore, we conclude that: 
\( BQ \perp AC \) and \( CP \perp AB \)

Now, consider triangle \( \triangle ABC \): 
The area can be calculated in two ways:

\[ \text{Area} = \frac{1}{2} \times AB \times CP = \frac{1}{2} \times AC \times BQ \]

Equating the two expressions: \[ \frac{1}{2} \times AB \times CP = \frac{1}{2} \times AC \times BQ \Rightarrow BQ = \frac{AB \times CP}{AC} \]

Substituting the values: \( AB = 30 \), \( CP = 20 \), \( AC = 25 \)

\[ BQ = \frac{30 \times 20}{25} = 24 \text{ cm} \]

Final Answer: \( \boxed{BQ = 24 \text{ cm}} \)