Question

In the triangle ABC, AB = 6, BC = 8 and AC = 10. A perpendicular dropped from B, meets the side AC at D. A circle of radius BD (with center B) is drawn. If the circle cuts AB and BC at P and Q respectively, the AP:QC is equal to

• A. 1:1
• B. 3:2
• C. 4:1
• D. 3:8

Hint:

Check whether the triangle is right angled. Then find the altitude from the right angle to the longest side, since that is the radius of the circle.

Answer 1

The Correct Option is D

Step 1: Understand the figure.
The sides of the triangle are 6, 8 and 10. Check \( 6^2 + 8^2 = 36 + 64 = 100 = 10^2 \). So the triangle is right angled at B, and AC = 10 is the hypotenuse.
BD is the perpendicular from B on the hypotenuse AC, so BD is the altitude to the hypotenuse.

Step 2: Find the radius BD.
Write the area of the triangle in two ways. Taking AB and BC as base and height gives \( \frac{1}{2} \times 6 \times 8 = 24 \). Taking AC as the base and BD as the height gives \( \frac{1}{2} \times 10 \times BD \).
Equating the two, \( 5 \times BD = 24 \), so \( BD = 4.8 \). The circle has centre B and radius 4.8.

Step 3: Locate P and Q.
The circle cuts AB at P, so \( BP = 4.8 \). Since \( 4.8 < 6 \), P lies between A and B.
The circle cuts BC at Q, so \( BQ = 4.8 \). Since \( 4.8 < 8 \), Q lies between B and C.

Step 4: Find AP and QC.
\[ AP = AB - BP = 6 - 4.8 = 1.2 \]
\[ QC = BC - BQ = 8 - 4.8 = 3.2 \]

Step 5: Take the ratio.
\[ AP : QC = 1.2 : 3.2 = 12 : 32 = 3 : 8 \]

Step 6: Why the other options fail.
Option 1 (1:1) would need AP = QC, but 1.2 is not 3.2.
Option 2 (3:2) would make AP larger than QC, but AP is the smaller one.
Option 3 (4:1) is wrong for the same reason, because AP is smaller than QC, not four times bigger.
Only 3:8 matches 1.2:3.2.

Final Answer:
The radius is 4.8, so AP is 1.2 and QC is 3.2. Their ratio is 3:8, which is option 4. \[ \boxed{4} \]