(B) In the figure given above, $\triangle ABC \sim \triangle XYZ$ , then find the values of x and y. ( $\triangle ABC$ : $AB = 4 \text{ cm}, BC = 6 \text{ cm}, AC = y$ ; $\triangle XYZ$ : $XY = x, YZ = 7.2 \text{ cm}, XZ = 6 \text{ cm}$ )

Q: 22. (B) In the figure given above, $\triangle ABC \sim \triangle XYZ$, then find the values of x and y. ($\triangle ABC$: $AB = 4 \text{ cm}, BC = 6 \text{ cm}, AC = y$; $\triangle XYZ$: $XY = x, YZ = 7.2 \text{ cm}, XZ = 6 \text{ cm}$)

$x = 4.8 \text{ cm}$ and $y = 5 \text{ cm}$. Since $\triangle ABC \sim \triangle XYZ$, corresponding sides give the scale factor $\frac{BC}{YZ} = \frac{6}{7.2} = \frac{5}{6}$, so $\frac{AB}{XY} = \frac{5}{6}$ gives $x = 4.8 \text{ cm}$ and $\frac{AC}{XZ} = \frac{5}{6}$ gives $y = 5 \text{ cm}$.

Q: How are x and y found from the scale factor?

Set each unknown pair equal to $\frac{5}{6}$: $\frac{4}{x} = \frac{5}{6} \text{ gives } x = \frac{24}{5} = 4.8 \text{ cm}$, and $\frac{y}{6} = \frac{5}{6} \text{ gives } y = 5 \text{ cm}$.

Triangles

Corresponding sides of similar triangles

Official Answer

$x = 4.8 \text{ cm}$ and $y = 5 \text{ cm}$ . Since $\triangle ABC \sim \triangle XYZ$ , corresponding sides give the scale factor $\frac{BC}{YZ} = \frac{6}{7.2} = \frac{5}{6}$ , so $\frac{AB}{XY} = \frac{5}{6}$ gives $x = 4.8 \text{ cm}$ and $\frac{AC}{XZ} = \frac{5}{6}$ gives $y = 5 \text{ cm}$ .

similar triangles corresponding sides△ABC ~ △XYZAB/XY = BC/YZ = AC/XZscale factor 5/6x = 4.8 cmy = 5 cmratio of sides

Marking Scheme

1½ mark: Writing the correct correspondence $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}$ .
2½ mark: Finding the scale factor $\frac{6}{7.2} = \frac{5}{6}$ .
3½ mark: Correct value $x = 4.8 \text{ cm}$ .
4½ mark: Correct value $y = 5 \text{ cm}$ (units required for full marks).

Hint

Write corresponding sides in letter order: $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}$ . First find the fully-known ratio $\frac{6}{7.2} = \frac{5}{6}$ — that scale factor gives both x and y.

Quick Oral Answer

Because $\triangle ABC \sim \triangle XYZ$ , corresponding sides are proportional in letter order, so $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}$ ; the known ratio $\frac{BC}{YZ} = \frac{6}{7.2} = \frac{5}{6}$ is the scale factor, giving $x = 4.8 \text{ cm}$ and $y = 5 \text{ cm}$ .

Analysis & Explanation

This question uses the definition of similar triangles, where corresponding sides follow the letter-order of the similarity statement.

Concept

$\triangle ABC \sim \triangle XYZ \Rightarrow \frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}$ ( $A \leftrightarrow X, B \leftrightarrow Y, C \leftrightarrow Z$ ).
Compute the scale factor first using the one ratio with both sides known: $\frac{BC}{YZ} = \frac{6}{7.2} = \frac{5}{6}$ .

Key points

Using $k = \frac{5}{6}$ : $\frac{4}{x} = \frac{5}{6} \Rightarrow x = 4.8 \text{ cm}$ ; $\frac{y}{6} = \frac{5}{6} \Rightarrow y = 5 \text{ cm}$ .
Verify all three ratios equal $\frac{5}{6}$ for consistency.

Common mistakes

Mismatching corresponding sides (e.g. pairing AB with YZ) because they 'look' similar in size instead of following the letter order.
Inverting the ratio (writing $\frac{x}{4} = \frac{5}{6}$ ) or mishandling the decimal 7.2 — converting it to 5/6 early avoids errors.

Common Mistakes

1Mismatching corresponding sides — e.g. pairing BC with XZ instead of YZ — because the letter order of the similarity statement was ignored.
2Inverting a ratio, writing $\frac{x}{4} = \frac{5}{6}$ instead of $\frac{4}{x} = \frac{5}{6}$ , which gives $x = \frac{10}{3}$ instead of 4.8.
3Dropping units or leaving x as an ugly decimal without simplifying $\frac{6}{7.2}$ to the exact fraction $\frac{5}{6}$ first, causing rounding errors.

Interesting Facts

The scale factor of similar figures affects area by its SQUARE: here $k = \frac{5}{6}$ , so $\frac{\text{area}(ABC)}{\text{area}(XYZ)} = \frac{25}{36}$ — a fact CBSE loves to combine with side-ratio questions.

Similarity is the mathematical basis of scale models, maps, and photographic enlargements: every point is scaled by the same factor while all angles are preserved.

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Frequently Asked Questions

How do I know which sides correspond in $\triangle ABC \sim \triangle XYZ$ ?

Follow the letter order of the similarity statement: $A \leftrightarrow X, B \leftrightarrow Y, C \leftrightarrow Z$ . So AB corresponds to XY, BC to YZ, and AC to XZ, giving $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}$ .

What is the scale factor in this problem?

Use the pair where both sides are known: $\frac{BC}{YZ} = \frac{6}{7.2} = \frac{5}{6}$ . This $\frac{5}{6}$ is the constant scale factor that every corresponding pair of sides must satisfy.

How are x and y found from the scale factor?

Set each unknown pair equal to $\frac{5}{6}$ : $\frac{4}{x} = \frac{5}{6} \text{ gives } x = \frac{24}{5} = 4.8 \text{ cm}$ , and $\frac{y}{6} = \frac{5}{6} \text{ gives } y = 5 \text{ cm}$ .