Question

In the given figure, \( \triangle AHK \sim \triangle ABC \). If \( AK = 10 \text{ cm} \), \( BC = 3.5 \text{ cm} \) and \( HK = 7 \text{ cm} \), find the length of \( AC \).

Hint:

Identify the pair of sides where both values are known to find the scale factor of similarity first.

Answer 1

Given:
△AHK ∼ △ABC
AK = 10 cm
BC = 3.5 cm
HK = 7 cm

Step 1: Use similarity ratio
From △AHK ∼ △ABC, the corresponding sides are proportional:
\[ \frac{HK}{BC} = \frac{AK}{AC} \]
Substitute the values:
\[ \frac{7}{3.5} = \frac{10}{AC} \]
Simplify left side:
\[ \frac{7}{3.5} = 2 \]
So, \[ 2 = \frac{10}{AC} \]
Cross multiply:
\[ AC = \frac{10}{2} = 5\ \text{cm} \]

Final Answer:
\[ \boxed{AC = 5\ \text{cm}} \]