Circles — Chapter Test | Class 10 IMO

Q1

From an external point P, OP = 10 cm and the radius is 6 cm. The length of the tangent from P is:

Tangent length = √(OP² − r²) = √(100 − 36) = 8 cm.

Q2

Which statement is true?

There is a tangent at every point of the circle, so there are infinitely many.

Q3

From an external point P at distance 2r from the centre of a circle of radius r, the angle between the two tangents is:

sin(half-angle) = r/(2r) = 1/2, so the half-angle is 30° and ∠APB = 60°.

Q4

A bicycle wheel of radius 35 cm touches the road. The road acts as a tangent, meeting the radius at the contact point at an angle of:

The road (tangent) is perpendicular to the radius at the contact point.

Q5

The tangent at any point of a circle is perpendicular to the:

It is perpendicular to the radius drawn to the point of contact.

Q6

The number of common tangents to two circles that touch each other externally is:

Externally touching circles have 3 common tangents.

Q7

PA and PB are tangents to a circle with centre O. If ∠AOB = 100°, then ∠APB equals:

∠AOB + ∠APB = 180°, so ∠APB = 80°.

Q8

From a point inside the circle, the number of tangents is 0 because:

Any line through an interior point meets the circle at two points, so it is a secant, not a tangent.

Q9

A secant differs from a tangent because a secant:

A secant intersects the circle in two points; a tangent in one.

Q10

The number of common tangents to two circles that lie apart (neither touching nor intersecting) is:

Two separate circles have 4 common tangents.

Q11

A circular coin of radius 1 cm rests touching a ruler's edge. The edge is a tangent, so the distance from the coin's centre to the edge is:

The distance from the centre to a tangent equals the radius, 1 cm.

Q12

The angle in a semicircle is a right angle; this means a diameter subtends at the circumference an angle of:

The angle in a semicircle is 90°.

Q13

Two straight roads are tangents drawn from a junction P to a circular park. A visitor notes the two tangent paths to the park are:

Tangents from an external point are equal, so the two paths are equal.

Q14

The number of tangents that can be drawn to a circle from an external point is:

Exactly two tangents can be drawn from a point outside a circle.

Q15

If OP = 13 cm and the radius is 5 cm, the length of each tangent from P is:

√(13² − 5²) = √144 = 12 cm.