Circles — Previous Year Questions (Class 10 Mathematics)
Circles are one of the most important geometric shapes with numerous properties related to tangents, chords, and angles. Understanding these properties is
TL;DR: Circles are one of the most important geometric shapes with numerous properties related to tangents, chords, and angles. Understanding these propertie…
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Circles are one of the most important geometric shapes with numerous properties related to tangents, chords, and angles. Understanding these properties is
Circles — Previous Year Questions with Solutions
Q (2023, 3 marks): A line is drawn from the center of a circle to a chord. Prove that this line bisects the chord if and only if it is perpendicular to the chord.
Answer: Let O be the center, AB be the chord, and OM be perpendicular to AB, where M is on AB.
In triangle OAM and OBM:
OA = OB (radii of the same circle)
OM = OM (common side)
Angle OMA = Angle OMB = 90° (given OM ⊥ AB)
By RHS criterion, triangle OAM ≅ triangle OBM
Therefore, AM = BM, which means M bisects AB.
Conversely, if M bisects AB, then by symmetry OM ⊥ AB.
Q (2022, 2 marks): Two circles have radii 5 cm and 3 cm. The distance between their centers is 10 cm. Determine whether the circles intersect, are tangent, or do not intersect.
Answer: Let r1 = 5 cm, r2 = 3 cm, and d = 10 cm (distance between centers).
For intersection: d < r1 + r2
10 < 5 + 3 = 8? No
For tangency: d = r1 + r2 or d = |r1 - r2|
10 = 8? No, and 10 = 2? No
For no intersection: d > r1 + r2
10 > 8? Yes
Therefore, the circles do not intersect.
Q (2023, 2 marks): Find the length of the tangent drawn from an external point P to a circle with center O and radius 6 cm, if the distance OP = 10 cm.
Answer: Let PT be the tangent from P to the circle, touching at point T.
OT ⊥ PT (radius perpendicular to tangent)
In right triangle OTP:
OP^2 = OT^2 + PT^2
10^2 = 6^2 + PT^2
100 = 36 + PT^2
PT^2 = 64
PT = 8 cm
Q (2022, 2 marks): The angle subtended by an arc at the center of a circle is 120°. Find the angle subtended by the same arc at a point on the major arc.
Answer: By the inscribed angle theorem, the angle subtended by an arc at a point on the circle is half the angle subtended at the center.
Angle at center = 120°
Angle at a point on the major arc = 120° / 2 = 60°
Q (2023, 3 marks): Two tangents are drawn to a circle from an external point. Prove that the line joining the external point and the center bisects the angle between the two tangents.
Answer: Let P be the external point, O be the center, and PA and PB be the two tangents touching the circle at A and B respectively.
In triangle OAP and OBP:
OA = OB (radii)
PA = PB (tangents from external point are equal)
OP = OP (common side)
By SSS criterion, triangle OAP ≅ triangle OBP
Therefore, angle APO = angle BPO
This means OP bisects angle APB.
Q (2021, 2 marks): A chord AB of a circle with center O subtends an angle of 60° at the center. If the radius of the circle is 8 cm, find the length of the chord AB.
Answer: In triangle OAB:
OA = OB = 8 cm (radii)
Angle AOB = 60°
Since OA = OB, triangle OAB is isosceles.
Drop a perpendicular from O to AB at point M.
Angle AOM = 30° (half of 60°)
In right triangle OAM:
AM = OA × sin(30°) = 8 × 1/2 = 4 cm
AB = 2 × AM = 2 × 4 = 8 cm
Frequently Asked Questions
What is the relationship between tangent and radius at the point of contact?
The radius drawn to the point of contact of a tangent is perpendicular to the tangent. This is a fundamental property used in many circle theorems and problems.
How many tangents can be drawn to a circle from an external point?
Exactly two tangents can be drawn from any external point to a circle. These tangents are equal in length and the line joining the external point to the center bisects the angle between them.
More Class 10 Mathematics PYQs
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