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A square, with sides of length x cm, is inside a circle - Edexcel - GCSE Maths - Question 8 - 2017 - Paper 3

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A square, with sides of length x cm, is inside a circle. Each vertex of the square is on the circumference of the circle. The area of the circle is 49 cm². Work ou... show full transcript

Worked Solution & Example Answer:A square, with sides of length x cm, is inside a circle - Edexcel - GCSE Maths - Question 8 - 2017 - Paper 3

Step 1

First, find the radius of the circle

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Answer

The area of the circle is given as 49 cm². The formula for the area of a circle is:

A = rac{22}{7} r^2

This means:

49 = rac{22}{7} r^2

To find the radius, rearranging gives:

r2=49⋅722r^2 = 49 \cdot \frac{7}{22}

Calculating this yields:

r2=34322≈15.5909r^2 = \frac{343}{22} \approx 15.5909

Therefore:

r≈15.5909≈3.95cmr \approx \sqrt{15.5909} \approx 3.95 cm

Step 2

Now apply the properties of the square and the circle

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Answer

Since the vertices of the square are on the circumference of the circle, the diagonal of the square equals the diameter of the circle. The relationship between the side length x of the square and its diagonal d is given by:

d=x2d = x\sqrt{2}

The diameter d of the circle can also be expressed as:

d=2r≈2⋅3.95≈7.9cmd = 2r \approx 2 \cdot 3.95 \approx 7.9 cm

Setting these two expressions for d equal gives:

x2=7.9x \sqrt{2} = 7.9

Thus, we can solve for x:

x=7.92≈5.585x = \frac{7.9}{\sqrt{2}} \approx 5.585

Step 3

Final Answer

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Answer

Rounding the answer to 3 significant figures, we get:

x≈5.59cmx \approx 5.59 cm

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