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, their corresponding sides are proportional. This property is used in the Basic Proportionality Theorem, where the ratios of the corresponding sides of the triangles are equal.
Step 2: Key Formula or Approach:
For two similar triangles \(\Delta AHK \sim \Delta ABC\), the ratio of the corresponding sides is given by: \[ \frac{AH}{AB} = \frac{HK}{BC} = \frac{AK}{AC} \] Step 3: Detailed Explanation:
Given values:
\(AK = 10 \text{ cm}\)
\(BC = 3.5 \text{ cm}\)
\(HK = 7 \text{ cm}\)
Using the equality of ratios: \[ \frac{HK}{BC} = \frac{AK}{AC} \] Substituting the known values: \[ \frac{7}{3.5} = \frac{10}{AC} \] Simplifying the left side: \[ 2 = \frac{10}{AC} \] Solving for \(AC\): \[ AC = \frac{10}{2} \] \[ AC = 5 \text{ cm} \] Step 4: Final Answer:
The length of \(AC\) is 5 cm.