Question 232 marksVery Short AnswerSection B

OR

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

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 cmx = 4.8\text{ cm} and y=5 cmy = 5\text{ cm}, found using the scale factor 56\frac{5}{6} from BCYZ=67.2\frac{BC}{YZ} = \frac{6}{7.2}, applied via ABXY\frac{AB}{XY} and ACXZ\frac{AC}{XZ}.

Answer keywords:similar trianglescorresponding sidesproportional sidesscale factorratio of similarityAAA similarity

Marking scheme

  1. 1.1 mark: correctly setting up ABXY=BCYZ=ACXZ\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ} using the correct vertex correspondence, and finding the scale factor 5/65/6.
  2. 2.1 mark: correct final values x=4.8 cmx = 4.8\text{ cm} and y=5 cmy = 5\text{ cm}.

Hint

Since ΔABC∼ΔXYZ\Delta ABC \sim \Delta XYZ, match vertices in order (A-X, B-Y, C-Z) to get ABXY=BCYZ=ACXZ\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 ΔABC∼ΔXYZ\Delta ABC \sim \Delta XYZ, corresponding sides are proportional: ABXY=BCYZ=ACXZ\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}. Using BCYZ=67.2=56\frac{BC}{YZ} = \frac{6}{7.2} = \frac{5}{6} as the scale factor, I get x=4.8 cmx = 4.8\text{ cm} from AB/XY, and y=5 cmy = 5\text{ cm} from AC/XZ.

Common mistakes

  1. 1.Mismatching corresponding sides, e.g., using ABYZ\frac{AB}{YZ} instead of ABXY\frac{AB}{XY}, by ignoring the strict vertex order in ΔABC∼ΔXYZ\Delta ABC \sim \Delta XYZ.
  2. 2.Confusing which side is x and which is y after computing the scale factor, leading to swapped answers.
  3. 3.Arithmetic errors in simplifying 67.2\frac{6}{7.2} to 5/6 (forgetting to multiply numerator and denominator by 10 first to clear the decimal: 60/72=5/660/72 = 5/6).

Analysis and explanation

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


Concept

  • ΔABC∼ΔXYZ\Delta ABC \sim \Delta XYZ means corresponding sides are proportional in the strict vertex order A↔X,B↔Y,C↔ZA\leftrightarrow X, B\leftrightarrow Y, C\leftrightarrow Z: ABXY=BCYZ=ACXZ\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}.

Key points

  • Using the pair with both lengths known, BCYZ=67.2=56\frac{BC}{YZ} = \frac{6}{7.2} = \frac{5}{6} gives the scale factor.
  • ABXY=56\frac{AB}{XY} = \frac{5}{6} ⟹ 4x=56\frac{4}{x} = \frac{5}{6} ⟹ x=4.8 cmx = 4.8\text{ cm}.
  • ACXZ=56\frac{AC}{XZ} = \frac{5}{6} ⟹ y6=56\frac{y}{6} = \frac{5}{6} ⟹ y=5 cmy = 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).

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 56\frac{5}{6} here means ΔABC\Delta ABC is smaller than ΔXYZ\Delta XYZ by that ratio; equivalently, ΔXYZ\Delta XYZ is an enlargement of ΔABC\Delta ABC by a factor of 65\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.

Question details

Q23 of 56 questions

Board
CBSE
Class
Class 10
Subject
Mathematics (Standard)
Year
2026
Section
Section B
Marks
2
Type
Very Short Answer
Chapter
Triangles
Topic
Similarity of Triangles
Bloom's level
apply
Syllabus topic
Unit 4: Geometry — Triangles: Similar triangles and ratio of corresponding sides
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

How do you know which sides correspond in ΔABC∼ΔXYZ\Delta ABC \sim \Delta XYZ?

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

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

Because both BC=6BC = 6 and YZ=7.2YZ = 7.2 are fully known numerical values, this pair directly gives the scale factor 56\frac{5}{6}, which can then be applied to find the unknowns x and y.