Two circles of equal radius are located such that each circle passes through the centre of the other circle (see the figure). Given that the radius of each circle is r units, find the perimeter of the shape formed by the two circles in terms of r units. (Ignore the dotted portions that lie within the circles.)
Step-by-step solution
To find: Total length of the two outer (red) arcs
Idea: Triangles ABC and ABD are equilateral, so the dotted arc of each circle subtends 120° at its centre. The red arc is the remaining 240°, which is 23 of the circle.
- AB = r (B is on circle A), AC = r (radius of circle A) and BC = r (radius of circle B). So triangle ABC is equilateral and ∠CAB = 60° = ∠CBA.1 mark
- In the same way triangle ABD is equilateral: ∠BAD = 60° = ∠ABD. So ∠CAD = 120° = ∠CBD.½ mark
- The dotted arc CD of circle A subtends 120° at A, so it is 120360 = 13 of that circle. The red arc is the other 23. The same holds for circle B.½ mark
- Perimeter = 2 × 23 × 2πr = 83πr units.1 mark
Check: 83πr ≈ 8.38r, less than two full circles (4πr ≈ 12.57r) and more than one (6.28r) ✓.
Answer to write in the exam
AB = AC = BC = r ⇒ △ABC equilateral ⇒ ∠CAB = 60°
Similarly ∠BAD = 60° ⇒ ∠CAD = 120° = ∠CBD
Dotted arc = 13 of circle ⇒ red arc = 23 of circle
Perimeter = 2 × 23 × 2πr
∴ Perimeter = 83πr units
Common mistakes that cost marks
- Taking ∠CAD as 60°. Both triangles ABC and ABD contribute 60° each, so it is 120°.
- Using 13 (the dotted part) instead of 23 (the outer part).
- Adding the dotted arcs too; the question says to ignore them.
How this can come in the exam
Two circles of radius 21 cm pass through each other’s centres. The perimeter of the outer boundary of the shape is (use π = 227)
- 88 cm
- 132 cm
- 176 cm
- 264 cm
Show answer
(C) 176 cm
83 × 227 × 21 = 8 × 22 = 176 cm.
Assertion (A): If two equal circles pass through each other’s centres, the common chord subtends 120° at each centre.
Reason (R): The two centres and one meeting point form an equilateral triangle.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
Each of the two equilateral triangles gives a 60° angle at the centre, so the chord subtends 60° + 60° = 120°. R explains A.
Try one yourself
Two circles of radius 7 cm pass through each other’s centres. Find the length of the boundary of the shape they form. (Use π = 227.)
Show answer
83 × 227 × 7 = 1763 ≈ 58.67 cm.
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