Question

In a triangle ABC, \( AB = 6 \) cm, \( BC = 8 \) cm and \( AC = 10 \) cm. A perpendicular BD is drawn from B to a point D on AC. Taking B as the centre and BD as the radius, a circle is drawn that cuts AB and BC at points E and F respectively. Find the ratio of the length of AE to that of CF.

• A. 3 : 5
• B. 3 : 8
• C. 4 : 7
• D. 3 : 7

Hint:

First check if triangle ABC is right angled, then find the altitude BD using its area, and use BE = BF = BD since they are radii of the same circle.

Answer 1

The Correct Option is B

Step 1: Check the type of triangle.
The sides are 6, 8 and 10. Since \( 6^2 + 8^2 = 36 + 64 = 100 = 10^2 \), triangle ABC is right angled at B, with AC as the hypotenuse.

Step 2: Find the area of triangle ABC.
Using the two legs AB and BC as base and height:
\[ \text{Area} = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times 6 \times 8 = 24 \text{ cm}^2 \]

Step 3: Use the altitude BD to write the area a second way.
The same area also equals \( \frac{1}{2} \times AC \times BD \), taking AC as the base and BD as the height.
\[ \frac{1}{2} \times 10 \times BD = 24 \Rightarrow BD = \frac{48}{10} = 4.8 \text{ cm} \]

Step 4: Use the circle to locate E and F.
The circle is centred at B with radius BD, and E lies on AB while F lies on BC, so BE and BF are also radii of the same circle.
\[ BE = BF = BD = 4.8 \text{ cm} \]

Step 5: Find AE and CF.
\[ AE = AB - BE = 6 - 4.8 = 1.2 \text{ cm} \]
\[ CF = BC - BF = 8 - 4.8 = 3.2 \text{ cm} \]

Step 6: Simplify the ratio.
\[ AE : CF = 1.2 : 3.2 = 12 : 32 = 3 : 8 \]

Final Answer:
The ratio of AE to CF is 3 : 8. \[ \boxed{3 : 8} \]