MathsCBSE Class 10 · Basic · Board exam 2025, set 430/4/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2025: question 38

Set in the CBSE Class 10 board exam of 2025, paper 430/4/1, 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-basedSome Applications of Trigonometry

Rahim and Nadeem are two friends whose plots are adjacent to each other. Rahim's son made a drawing of the plots with necessary details. It is decided that Rahim will fence the triangular plot ABC and Nadeem will fence along the sides AF, FE and BE. Observe the diagram carefully and answer the following questions. (Use $2 = 1.41$ and $3 = 1.73$ )

(i)
Find length BC.
1 mark
(ii)
Find length AG.
1 mark
(iii)(a)
Calculate perimeter of $Δ$ ABC.
2 marks
OR
(iii)(b)
Calculate length of (AF + FE + EB).
2 marks

Figure in the paper: Plan of plots. Points A, G, D, H, B lie on one horizontal line (A left, B right). C is above D, with angle C = 60° and angle at A = 45°, angle at B = 30° (and 45° marked near B on the upper side). F is below G, with AF = 53.58 m, FE = 53 m, and E lies between F and H/B. Lengths marked: BH = 40 m, HD = 10 m, DG = 40 m, GF = 38 m, FE = 53 m, AF = 53.58 m. Vertical segments CD and HE are drawn; GF is vertical with a right-angle mark at G; DH is 10 m, HE is vertical. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/4/1.

Hints

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

Hint 1

Look for the right-angled triangle that contains the side you want and an angle you know (45°, 30° or 60°).

Hint 2

For BC use triangle CDB with DB = 10 + 40. For AC use triangle ADC with AD = 38 + 40.

Hint 3

Choose the ratio that links the known side to the unknown side. For example, adjacent/hypotenuse is cos.

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) $D B = D H + H B = 10 + 40 = 50$ m. In right $△ C D B$ , $∠ C B D = 4 5^{\circ}$ , so $cos 4 5^{\circ} = \frac{D B}{B C} \Rightarrow \frac{1}{2} = \frac{50}{B C} \Rightarrow B C = 502 = 50 \times 1.41 = 70.5$ m.
1 mark
(ii) In right $△ A GF$ , $∠ F A G = 4 5^{\circ}$ , so $tan 4 5^{\circ} = \frac{GF}{A G} \Rightarrow 1 = \frac{38}{A G} \Rightarrow A G = 38$ m.
1 mark
(iii)(a) $A D = A G + G D = 38 + 40 = 78$ m. In right $△ A D C$ , $∠ A C D = 6 0^{\circ}$ , so $sin 6 0^{\circ} = \frac{A D}{A C} \Rightarrow \frac{3}{2} = \frac{78}{A C} \Rightarrow A C = \frac{156}{3} = 523 = 52 \times 1.73 = 89.96$ m.
1 mark
(iii)(a) $A B = A G + G D + D H + H B = 38 + 40 + 10 + 40 = 128$ m. Perimeter of $△ A B C = A B + B C + C A = 128 + 70.5 + 89.96 = 288.46$ m.
1 mark
(iii)(b) In right $△ E H B$ , $∠ H B E = 3 0^{\circ}$ and $B H = 40$ m, so $cos 3 0^{\circ} = \frac{B H}{B E} \Rightarrow \frac{3}{2} = \frac{40}{B E} \Rightarrow B E = \frac{80}{3} = \frac{80 3}{3} = \frac{80 \times 1.73}{3} = 46.13$ m.
1 mark
(iii)(b) $A F + F E + E B = 53.58 + 53 + 46.13 = 152.71$ m.
1 mark

Answer: (i) BC = 70.5 m; (ii) AG = 38 m; (iii)(a) 288.46 m; (iii)(b) 152.71 m

Where the marks are

Uses $cos 4 5^{\circ} = \frac{50}{B C}$ to get BC = 70.5 m (1 mark)
Uses $tan 4 5^{\circ} = \frac{38}{A G}$ to get AG = 38 m (1 mark)
Uses $sin 6 0^{\circ} = \frac{78}{A C}$ to get AC = 89.96 m (1 mark)
Adds the sides to get perimeter 288.46 m (1 mark)
Uses $cos 3 0^{\circ} = \frac{40}{B E}$ to get BE = 46.13 m (1 mark)
Adds to get AF + FE + EB = 152.71 m (1 mark)

Revise the chapter

Chapter: Some Applications of Trigonometry (notes, formulas, where marks are lost and every past question)
Topic: Line of Sight and Angle of Elevation

Practise this chapterBoard-style questions from Some Applications of Trigonometry, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 430/4/1 (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 2025: question 38

Hints

Step-by-step solution

Revise the chapter

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