Question

In the figure given above, \(\triangle ABC \sim \triangle XYZ\), then find the values of \(x\) and \(y\).

Hint:

When cross-multiplying in BPT problems, look for algebraic identities like \((x+2)(x-2) = x^2-4\) to simplify your work.

Answer 1

Given:
ΔABC ∼ ΔXYZ
AB = 4 cm, BC = 6 cm, AC = y
XY = x, YZ = 7.2 cm, XZ = 6 cm

Step 1: Use similarity ratio
Since ΔABC ∼ ΔXYZ, the corresponding sides are proportional:
AB ↔ XY
BC ↔ YZ
AC ↔ XZ

So, \[ \frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ} \]

Step 2: Find the ratio using BC and YZ
\[ \frac{BC}{YZ} = \frac{6}{7.2} \] Simplify: \[ \frac{6}{7.2} = \frac{60}{72} = \frac{5}{6} \] So, similarity ratio = 5 : 6

Step 3: Find XY = x
\[ \frac{AB}{XY} = \frac{5}{6} \] \[ \frac{4}{x} = \frac{5}{6} \] Cross-multiply: \[ 5x = 24 \] \[ x = \frac{24}{5} = 4.8\ \text{cm} \]

Step 4: Find AC = y
\[ \frac{AC}{XZ} = \frac{5}{6} \] \[ \frac{y}{6} = \frac{5}{6} \] \[ y = 5\ \text{cm} \]

Final Answers:
\(x = 4.8\ \text{cm}\)
\(y = 5\ \text{cm}\)