MathsCBSE Class 10 · Basic · Board exam 2024, set 430/3/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2024: question 38

Set in the CBSE Class 10 board exam of 2024, paper 430/3/1, section E, question 38, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q38 · 4 marks · Case-basedCompetency-basedTriangles

In the figure given below, a folding table is shown. The legs of the table are represented by line segments $A B$ and $C D$ intersecting at $O$ . Join $A C$ and $B D$ . Considering table top is parallel to the ground, and $O B = x$ , $O D = x + 3$ , $O C = 3 x + 19$ and $O A = 3 x + 4$ , answer the following questions:

(i)
Prove that $△ O A C$ is similar to $△ O B D$ .
1 mark
(ii)
Prove that $\frac{O A}{A C} = \frac{O B}{B D}$ .
1 mark
(iii)
(a) Observe the figure and find the value of $x$ . Hence, find the length of $O C$ .
2 marks
OR
(iii)
(b) Observe the figure and find $\frac{B D}{A C}$ .
2 marks

Figure in the paper: A bow-tie shaped figure formed by two line segments AB and CD intersecting at O. D is at the upper left, B at the upper right, A at the lower left and C at the lower right. Segments AC and BD are joined forming the two triangles OAC and OBD. Labels: OD = x + 3, OB = x, OA = 3x + 4, OC = 3x + 19. The top edge DB is horizontal, parallel to the ground line AC. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/3/1.

Hints

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

Hint 1

Since the table top is parallel to the ground, $D B ∥ A C$ ; look for alternate angles with transversals AB and CD.

Hint 2

Match the vertices correctly: O↔O, A↔B, C↔D, then write the ratios of corresponding sides.

Hint 3

Equate $\frac{O A}{O B}$ and $\frac{O C}{O D}$ and cross-multiply; the $x^{2}$ terms cancel.

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.

(i) $D B ∥ A C$ because the table top is parallel to the ground.
In $△ O A C$ and $△ O B D$ :
- $∠ O A C = ∠ O B D$ [alternate interior angles]
- $∠ O C A = ∠ O D B$ [alternate interior angles]
$∴ △ O A C \sim △ O B D$ [AA similarity]
1 mark
(ii) Since $△ O A C \sim △ O B D$ , corresponding sides are proportional:
$\frac{O A}{O B} = \frac{A C}{B D}$
Rearranging gives $\frac{O A}{A C} = \frac{O B}{B D}$ .
1 mark
(iii)(a) From the similarity, $\frac{O A}{O B} = \frac{O C}{O D}$ , so $\frac{3 x + 4}{x} = \frac{3 x + 19}{x + 3}$ .
Cross-multiplying: $(3 x + 4) (x + 3) = x (3 x + 19)$
$3 x^{2} + 13 x + 12 = 3 x^{2} + 19 x$
$6 x = 12 \Rightarrow x = 2$
1½ marks
(iii)(a) $O C = 3 (2) + 19 = 25$ units
½ mark
(iii)(b) From the similarity, $\frac{O B}{O A} = \frac{O D}{O C} = \frac{B D}{A C}$ , so $\frac{x}{3 x + 4} = \frac{x + 3}{3 x + 19}$ .
Cross-multiplying: $x (3 x + 19) = (x + 3) (3 x + 4)$ , which gives $6 x = 12$ , so $x = 2$ .
1½ marks
(iii)(b) $\frac{B D}{A C} = \frac{O B}{O A} = \frac{2}{10} = \frac{1}{5}$
½ mark

Answer: (i) △OAC ~ △OBD by AA; (ii) OA/AC = OB/BD proved; (iii)(a) x = 2, OC = 25; (iii)(b) BD/AC = 1/5

Where the marks are

(i) Two pairs of equal alternate angles (from $D B ∥ A C$ ) and the AA similarity conclusion (1 mark)
(ii) Uses proportional corresponding sides $\frac{O A}{O B} = \frac{A C}{B D}$ and rearranges (1 mark)
(iii)(a) Sets up $\frac{3 x + 4}{x} = \frac{3 x + 19}{x + 3}$ and solves $x = 2$ (1½ marks)
(iii)(a) $O C = 25$ (½ mark)
(iii)(b) Sets up the proportion and gets $x = 2$ (1½ marks)
(iii)(b) $\frac{B D}{A C} = \frac{1}{5}$ (½ mark)

Revise the chapter

Chapter: Triangles (notes, formulas, where marks are lost and every past question)
Topic: Problems on Similar Triangles

Practise this chapterBoard-style questions from Triangles, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 430/3/1 (Basic): 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 Basic 2024: question 38

Hints

Step-by-step solution

Revise the chapter

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