The sum of areas of two squares is 2650\ cm². If the sum of their perimeters is 280 cm, find the sides of the two given squares.

MathsCBSE Class 10 · Basic · Board exam 2025, set 430/2/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2025: question 32(a)

Set in the CBSE Class 10 board exam of 2025, paper 430/2/1, section D, question 32(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(a) · 5 marks · Long answerQuadratic Equations

The sum of areas of two squares is $2650 c m^{2}$ . If the sum of their perimeters is 280 cm, find the sides of the two given squares.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/2/1.

Hints

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

Hint 1

Use two unknowns for the sides and write one equation for areas and one for perimeters.

Hint 2

Express one side in terms of the other from the perimeter equation and substitute.

Hint 3

Factorise the quadratic: look for two numbers with product 1125 and sum 70.

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.

Let the sides of the squares be $x$ cm and $y$ cm. Sum of areas: $x^{2} + y^{2} = 2650$ ...(i)
1 mark
Sum of perimeters: $4 x + 4 y = 280 \Rightarrow x + y = 70 \Rightarrow y = 70 - x$ ...(ii)
1 mark
Substituting in (i): $x^{2} + (70 - x)^{2} = 2650 \Rightarrow 2 x^{2} - 140 x + 4900 - 2650 = 0 \Rightarrow 2 x^{2} - 140 x + 2250 = 0$ , i.e. $x^{2} - 70 x + 1125 = 0$ .
1 mark
$x^{2} - 25 x - 45 x + 1125 = 0 \Rightarrow (x - 25) (x - 45) = 0 \Rightarrow x = 25$ or $x = 45$ .
1 mark
Then $y = 45$ or $y = 25$ . So the sides of the squares are 25 cm and 45 cm.
1 mark

Answer: 25 cm and 45 cm

Where the marks are

Forms $x^{2} + y^{2} = 2650$ (1 mark)
Forms $x + y = 70$ (1 mark)
Obtains quadratic $x^{2} - 70 x + 1125 = 0$ (1 mark)
Solves to get $x = 25, 45$ (1 mark)
Gets corresponding y and states sides 25 cm and 45 cm (1 mark)

Revise the chapter

Chapter: Quadratic Equations (notes, formulas, where marks are lost and every past question)
Topic: Word Problems on Quadratic Equations

Practise this chapterBoard-style questions from Quadratic Equations, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 430/2/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2025: question 32(a)

Hints

Step-by-step solution

Revise the chapter

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