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

Step 1: Understanding the Concept:
In similar triangles, the ratios of the lengths of corresponding sides are equal.
Step 2: Key Formula or Approach:
Given \( \triangle AHK \sim \triangle ABC \), the corresponding sides are:
\[ \frac{AH}{AB} = \frac{HK}{BC} = \frac{AK}{AC} \]
Step 3: Detailed Explanation:
From the given data: \( AK = 10 \text{ cm} \), \( BC = 3.5 \text{ cm} \), \( HK = 7 \text{ cm} \).
We use the ratio:
\[ \frac{HK}{BC} = \frac{AK}{AC} \]
Substituting the values:
\[ \frac{7}{3.5} = \frac{10}{AC} \]
Since \( 3.5 \times 2 = 7 \):
\[ 2 = \frac{10}{AC} \]
\[ AC = \frac{10}{2} = 5 \text{ cm} \]
Step 4: Final Answer:
The length of AC is 5 cm.