Two circles of equal radius are located such that each circle passes through the centre Class 9

Two circles of equal radius are located such that each circle passes through the centre Class 9

Question 1.

Two circles of equal radius are located such that each circle passes through the centre of the other circle. Class 9

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.)

Answer:

Given, two congruent circles with centres A and B.

Each circle passes through the centre of the other.

Join A to B, A to C, and B to C to form ∆ ABC and join A to D, and B to D to form ∆ ABD

Since AB = r, AC = r, BC = r

∴ ∆ABC is equilateral triangle

∴ ∠CAB = ∠CBA = 60° ..

Similarly, ∠BAD = ∠ABD = 60°

∴ ∠CAD = ∠CBD = 120°

Since one-third angle of whole circle = 120°.

Thus, each dotted arc is one-third of the circumference of the circle on which it lies. Hence, each required (red) arc is two-third of the circumference.

Total length of two red arcs

$=2 \times \frac{2}{3} \times 2 \pi r=\frac{8}{3} \pi r$