Case Study: There is a circular park in a colony. The diameter of the park is 40 m. Three poles A, B and C have been eracted at equal distances on the boundary of the park. These poles have been connected with each other using straight wires, as shown in the figure.

Based on this information, answer the following questions:

1. The circumference of the circular park is :
(a) 125.6 m
(b) 130 m
(c) 251.2 m
(d) 257.2 m

2. A child cycles along the boundary of the park in clockwise direction. The distance covered by the child in going from B to C is :
(a) 41.8 m
(b) 83.7 m
(c) 20.9 m
(d) 125.6 m

3. ABC is :
(a) a scalene triangle
(b) a right triangle
(c) an equilateral triangle
(d) an isosceles triangle

4. If we draw perpendicular from B on AC, then it will pass through : (a) centre of the circle
(b) circumcentre of the circle
(c) centroid of the circle
(d) all the above

5. The length of the piece of wire used to connect any two poles is :
(a) 20 m
(b) 20$\sqrt{3}$ m
(c) 20$\sqrt{2}$ m
(d) 60$\sqrt{3}$ m

1. Given,

d = 40 m

r = $\dfrac{40}{2}$ = 20 m

The circumference of the circular park is = 2πr

= 2 × $\dfrac{22}{7}$ × 20

= 125.6 m

Hence, option (a) is the correct option.

2. Since poles A, B, and C are at equal distances on the boundary, they divide the total circumference into three equal arcs.

The distance along the boundary (arc BC) = $\dfrac{\text{Circumference}}{3} = \dfrac{125.6}{3}$ = 41.8 m

From the figure, going clockwise from B to C means the child travels the major arc, not the minor arc.

Major arc = Total circumference − minor arc

= 125.6 − 41.87 = 83.73 m

Hence, option (b) is the correct option.

3. Poles A, B, and C are at equal distances on the boundary.

Since, arc BA = arc AC = arc CB.

This means the chords AB, BC, and CA are equal in length.

A triangle with three equal sides is an equilateral triangle.

Hence, option (c) is the correct option.

4. In an equilateral triangle, the perpendicular bisector, median, and altitude are all the same line.

Thus, perpendicular from B on AC, will pass through the center, centroid and circumcenter of the circle.

Hence, option (d) is the correct option.

5. Radius of inscribed circle = 20 m

The relationship between the side of an equilateral triangle and the circumradius is:

Side length = $r\sqrt3 = 20\sqrt3.$

Hence, option (b) is the correct option.