Circles — Chapter Test | Class 9 IMO

Q1

If ABCD is a cyclic quadrilateral and ∠A = 70°, what is the measure of ∠C?

The sum of opposite angles in a cyclic quadrilateral is 180°. Therefore, ∠C = 180° − 70° = 110°.

Q2

In a Venn diagram, circle A represents students playing Cricket, circle B represents Football, and circle C represents Hockey. What does the region common to all three circles represent?

In set theory and Venn diagrams, the intersection of all three sets represents the elements that belong to all categories simultaneously.

Q3

If a regular hexagon is inscribed in a circle of radius r, what is the perimeter of the hexagon?

A regular hexagon can be divided into 6 equilateral triangles from the centre. Each side of the hexagon is equal to the radius r. Perimeter = 6 × side = 6r.

Q4

A circular cycling track has an inner radius of 21 m and an outer radius of 28 m. What is the width of the track?

The width of the circular track is the difference between the outer radius and the inner radius. Width = 28 m − 21 m = 7 m.

Q5

A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc.

The chord forms an equilateral triangle with the radii, so the angle at the centre is 60°. The angle at the major arc is 60°/2 = 30°. For the cyclic quad formed by the chord, major arc point, and minor arc point, the angle at the minor arc is 180° − 30° = 150°.

Q6

In a circle with centre O, a chord AB has length 8 cm. If the distance from O to AB is 3 cm, what is the radius of the circle?

The perpendicular from the centre bisects the chord. So, half of chord AB = 4 cm. Using Pythagoras theorem: radius² = 3² + 4² = 9 + 16 = 25. Radius = √25 = 5 cm.

Q7

Two circles with radii 5 cm and 3 cm intersect at two points. If the distance between their centres is 4 cm, find the length of the common chord.

The sides of the triangle formed by the two radii and the distance between centres are 3, 4, 5. This forms a right-angled triangle at the centre of the smaller circle. The common chord is perpendicular to the line of centres. The smaller circle's radius acts as half the common chord because it's the altitude. Thus, common chord = 2 × 3 = 6 cm.

Q8

In a logical sequence of figures, the first circle is divided into 2 sectors, the second into 4 sectors, and the third into 8 sectors. Following this pattern, how many sectors will the fifth circle have?

The number of sectors follows the pattern 2¹, 2², 2³. The fourth circle will have 2⁴ = 16. The fifth circle will have 2⁵ = 32.

Q9

How many total lines of symmetry does a perfect circle have?

Any straight line passing through the centre of a circle divides it into two identical semicircles. Since there are infinite such lines (diameters), a circle has infinite lines of symmetry.

Q10

The angle in a semicircle is a:

The angle subtended by a diameter at any point on the circumference is always 90°, making it a right angle.

Q11

Meena makes a circular rangoli design in her courtyard. She wants to divide the boundary into equal arcs using 8 dots. What will be the measure of the angle subtended by each arc at the centre?

Total angle at the centre is 360°. Since there are 8 equal arcs, each arc subtends 360° / 8 = 45°.

Q12

At 3:30, what is the angle between the hour and minute hands of a clock? (Assuming a standard circular clock face)

At 3:30, the minute hand is on 6. The hour hand has moved halfway between 3 and 4. The angle between 3 and 6 is 90°. The hour hand moves 30°/2 = 15° towards 4. So the angle is 90° − 15° = 75°.

Q13

The wheel of an Indian school bus has a diameter of 1.4 m. How many complete revolutions will the wheel make to cover a distance of 4.4 km?

Radius = 0.7 m. Circumference = 2 × 22/7 × 0.7 = 4.4 m. Distance = 4.4 km = 4400 m. Number of revolutions = Total Distance / Circumference = 4400 / 4.4 = 1000.

Q14

Two parallel chords of lengths 10 cm and 24 cm are on opposite sides of the centre of a circle of radius 13 cm. What is the distance between these chords?

For the 10 cm chord, half is 5 cm. Distance from centre = √(13² − 5²) = 12 cm. For the 24 cm chord, half is 12 cm. Distance from centre = √(13² − 12²) = 5 cm. Since they are on opposite sides, total distance = 12 + 5 = 17 cm.

Q15

The angle subtended by an arc at the centre is 80°. What is the angle subtended by the same arc at any point on the remaining part of the circle?

The angle subtended by an arc at the centre is double the angle subtended by it at any remaining part of the circle. Therefore, 80° / 2 = 40°.