OR

In the figure given above, $\Delta ABC \sim \Delta XYZ$ , then find the values of x and y.

Q: OR In the figure given above, $\Delta ABC \sim \Delta XYZ$, then find the values of x and y.

$x = 4.8\text{ cm}$ and $y = 5\text{ cm}$, found using the scale factor $\frac{5}{6}$ from $\frac{BC}{YZ} = \frac{6}{7.2}$, applied via $\frac{AB}{XY}$ and $\frac{AC}{XZ}$.

Triangle ABC with A at top, B bottom-left, C bottom-right; side AB = 4 cm, side AC = y, side BC = 6 cm. Triangle XYZ wit — Fig. for Q23

Triangles

Similarity of Triangles

Official Answer

$x = 4.8\text{ cm}$ and $y = 5\text{ cm}$ , found using the scale factor $\frac{5}{6}$ from $\frac{BC}{YZ} = \frac{6}{7.2}$ , applied via $\frac{AB}{XY}$ and $\frac{AC}{XZ}$ .

similar trianglescorresponding sidesproportional sidesscale factorratio of similarityAAA similarity

Marking Scheme

11 mark: correctly setting up $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}$ using the correct vertex correspondence, and finding the scale factor $5/6$ .
21 mark: correct final values $x = 4.8\text{ cm}$ and $y = 5\text{ cm}$ .

Hint

Since $\Delta ABC \sim \Delta XYZ$ , match vertices in order (A-X, B-Y, C-Z) to get $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}$ , then use the fully known pair BC and YZ to find the ratio.

Quick Oral Answer

Since $\Delta ABC \sim \Delta XYZ$ , corresponding sides are proportional: $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}$ . Using $\frac{BC}{YZ} = \frac{6}{7.2} = \frac{5}{6}$ as the scale factor, I get $x = 4.8\text{ cm}$ from AB/XY, and $y = 5\text{ cm}$ from AC/XZ.

Analysis & Explanation

OR alternative to Q22 — tests proportional sides via triangle similarity instead of BPT.

Concept

$\Delta ABC \sim \Delta XYZ$ means corresponding sides are proportional in the strict vertex order $A\leftrightarrow X, B\leftrightarrow Y, C\leftrightarrow Z$ : $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}$ .

Key points

Using the pair with both lengths known, $\frac{BC}{YZ} = \frac{6}{7.2} = \frac{5}{6}$ gives the scale factor.
$\frac{AB}{XY} = \frac{5}{6}$ ⟹ $\frac{4}{x} = \frac{5}{6}$ ⟹ $x = 4.8\text{ cm}$ .
$\frac{AC}{XZ} = \frac{5}{6}$ ⟹ $\frac{y}{6} = \frac{5}{6}$ ⟹ $y = 5\text{ cm}$ .

Common mistakes

Matching sides by apparent size in the figure instead of by the stated correspondence order (A-X, B-Y, C-Z).

Common Mistakes

1Mismatching corresponding sides, e.g., using $\frac{AB}{YZ}$ instead of $\frac{AB}{XY}$ , by ignoring the strict vertex order in $\Delta ABC \sim \Delta XYZ$ .
2Confusing which side is x and which is y after computing the scale factor, leading to swapped answers.
3Arithmetic errors in simplifying $\frac{6}{7.2}$ to 5/6 (forgetting to multiply numerator and denominator by 10 first to clear the decimal: $60/72 = 5/6$ ).

Interesting Facts

In similar triangles, not only are corresponding sides proportional, but corresponding angles are equal, corresponding medians/altitudes/angle-bisectors are also in the same ratio, and areas are in the ratio of the square of the scale factor.

The scale factor $\frac{5}{6}$ here means $\Delta ABC$ is smaller than $\Delta XYZ$ by that ratio; equivalently, $\Delta XYZ$ is an enlargement of $\Delta ABC$ by a factor of $\frac{6}{5}$ .

Triangle similarity by SSS (all three sides proportional) is one of three main similarity criteria in the CBSE syllabus, alongside AA and SAS similarity.

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Frequently Asked Questions

How do you know which sides correspond in $\Delta ABC \sim \Delta XYZ$ ?

The order of vertices in the similarity statement fixes the correspondence: $A\leftrightarrow X, B\leftrightarrow Y, C\leftrightarrow Z$ , so $AB\leftrightarrow XY, BC\leftrightarrow YZ$ , and $CA\leftrightarrow ZX$ .

Why use $\frac{BC}{YZ}$ first to find the scale factor?

Because both $BC = 6$ and $YZ = 7.2$ are fully known numerical values, this pair directly gives the scale factor $\frac{5}{6}$ , which can then be applied to find the unknowns x and y.