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: Note the right angle, as before.
Since $6^2 + 8^2 = 10^2$, angle B is $90^{\circ}$, so AC is the hypotenuse of right triangle ABC.

Step 2: Write the sine and cosine of angle A.
In right triangle ABC, angle A is opposite side BC and adjacent to side AB, with hypotenuse AC.
$$\sin A = \frac{BC}{AC} = \frac{8}{10} = 0.8, \qquad \cos A = \frac{AB}{AC} = \frac{6}{10} = 0.6$$

Step 3: Use right triangle ABD to find BD with trigonometry.
In right triangle ABD (right angle at D), angle A is the same angle as in triangle ABC. So:
$$BD = AB \times \sin A = 6 \times 0.8 = 4.8 \text{ cm}$$

Step 4: Locate E and F using the radius.
The circle centred at B with radius BD = 4.8 cm meets AB at E and BC at F, so $BE = BF = 4.8$ cm, since both are radii of the same circle.

Step 5: Find AE and CF and simplify the ratio.
$$AE = AB - BE = 6 - 4.8 = 1.2 \text{ cm}, \qquad CF = BC - BF = 8 - 4.8 = 3.2 \text{ cm}$$
$$AE : CF = 1.2 : 3.2 = 3 : 8$$

Final Answer:
The ratio of AE to CF is 3 : 8, the same result reached here using the sine of angle A instead of the triangle's area. \[ \boxed{3 : 8} \]