MathsCBSE Class 10 · Standard · Board exam 2023, set 30/1/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Standard 2023: question 22

Set in the CBSE Class 10 board exam of 2023, paper 30/1/1, section B, question 22, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q22 · 2 marks · Very short answerTriangles

In the given figure, $X Z ∥ B C$ . $A Z = 3$ cm, $Z C = 2$ cm, $B M = 3$ cm and $M C = 5$ cm. Find the length of $X Y$ .

Figure in the paper: Triangle ABC with apex A at top, base BC horizontal (B left, C right). Point M lies on BC with BM = 3 cm and MC = 5 cm. Segment AM is drawn. X lies on AB and Z lies on AC, with XZ parallel to BC. Y is the point where XZ meets AM. Labels: AZ = 3 cm, ZC = 2 cm. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Since $X Z ∥ B C$ , use the BPT on $△ A B C$ to compare AX and XB.

Hint 2

Look for a triangle containing XY that is similar to $△ A B M$ , then use $A X / A B$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

In $△ A B C$ , $X Z ∥ B C$ , so by the Basic Proportionality Theorem $\frac{A X}{X B} = \frac{A Z}{Z C} = \frac{3}{2}$ .
½ mark
In $△ A X Y$ and $△ A B M$ , $∠ A$ is common and $∠ A X Y = ∠ A B M$ (corresponding angles, $X Y ∥ B M$ ). So $△ A X Y \sim △ A B M$ (AA similarity).
½ mark
Hence $\frac{X Y}{B M} = \frac{A X}{A B}$ . From the first step, $\frac{A X}{A B} = \frac{3}{3 + 2} = \frac{3}{5}$ , so $\frac{X Y}{3} = \frac{3}{5}$ .
½ mark
$X Y = \frac{9}{5} = 1.8$ cm.
½ mark

Answer: XY = 9/5 cm = 1.8 cm

Where the marks are

Uses BPT to get $A X / X B = 3/2$ (½ mark)
Proves $△ A X Y \sim △ A B M$ (½ mark)
Writes $X Y / B M = A X / A B = 3/5$ (½ mark)
Gets $X Y = 1.8$ cm (½ mark)

Revise the chapter

Chapter: Triangles (notes, formulas, where marks are lost and every past question)
Topic: Basic Proportionality Theorem (Thales Theorem)

Practise this chapterBoard-style questions from Triangles, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 30/1/1 (Standard): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Standard 2023: question 22

Hints

Step-by-step solution

Revise the chapter

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