Question

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

Hint:

To identify corresponding sides correctly, look at the order of vertices in the similarity statement: \(A \to A\), \(H \to B\), and \(K \to C\). Thus, \(HK\) corresponds to \(BC\) and \(AK\) corresponds to \(AC\).

Answer 1

Step 1: Understanding the Concept:
When two triangles are similar, the ratios of their corresponding sides are equal. This is known as the Basic Proportionality Theorem application in similarity.
Step 2: Key Formula or Approach:
For \(\Delta AHK \sim \Delta ABC\), the ratio of corresponding sides is:
\[ \frac{AH}{AB} = \frac{HK}{BC} = \frac{AK}{AC} \]
Step 3: Detailed Explanation:
Given values:
\(AK = 10 \text{ cm}\)
\(BC = 3.5 \text{ cm}\)
\(HK = 7 \text{ cm}\)
Using the equality of ratios:
\[ \frac{HK}{BC} = \frac{AK}{AC} \]
Substitute the known values:
\[ \frac{7}{3.5} = \frac{10}{AC} \]
Simplify the left side:
\[ 2 = \frac{10}{AC} \]
Solve for \(AC\):
\[ AC = \frac{10}{2} \]
\[ AC = 5 \text{ cm} \]
Step 4: Final Answer:
The length of \(AC\) is 5 cm.