Class 10 Circles (Tangents) — practice questions with answers
Geometry · 5 sample questions from the school syllabus. Try each one, then open the answer and the worked solution.
1. What is a tangent to a circle?
AA line that intersects the circle at two points
BA line that touches the circle at exactly one point
CA line that passes through the center of the circle
DA line that is parallel to the circle
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Answer: B) A line that touches the circle at exactly one point
A tangent is defined as a line that touches a circle at exactly one point.
2. Two tangents are drawn from an external point P to a circle with centre O. If angle between the two tangents is 60 degrees, find the angle subtended by the line segment joining the points of contact at the centre.
A60 degrees
B120 degrees
C90 degrees
D30 degrees
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Answer: B) 120 degrees
Using the quadrilateral OAPB, angle AOB = 180 - angle APB = 180 - 60 = 120 degrees (since the two right angles at A and B sum to 180 and the four angles total 360).
3. In the figure, PA and PB are tangents from P to a circle with centre O. If OA = 6 cm and OP = 10 cm, find the length of tangent PA.
A7
B8
C16
D9
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Answer: B) 8
PA² = OP² - OA² = 10² - 6² = 100 - 36 = 64, so PA = 8 cm.
4. Two tangents are drawn to a circle of radius 6 cm from an external point P, and the angle between the two tangents measures 60 degrees. What is the distance of P from the centre O of the circle?
A6 cm
B12 cm
C63 cm
D18 cm
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Answer: B) 12 cm
The line OP bisects the angle between the tangents, so the half-angle at P is 30 degrees. In the right triangle formed by the radius and the tangent, sin30∘=OP6, so OP=0.56=12 cm. (The value 63 is the tangent length, not OP.)
5. A circular table of radius 60 cm is pushed across a room until its edge just meets a straight flat wall at a single point. Which statement correctly describes the situation at that point of contact?
AThe wall acts as a chord of the circular edge of the table
BThe wall acts as a tangent to the circular edge, and the radius drawn to the point of contact is perpendicular to the wall
CThe wall acts as a secant of the circular edge and cuts it at two points
DThe wall acts as a tangent, and the radius drawn to the point of contact makes an angle of 60 degrees with the wall
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Answer: B) The wall acts as a tangent to the circular edge, and the radius drawn to the point of contact is perpendicular to the wall
The wall meets the circular edge at exactly one point, which is precisely the definition of a tangent. By the tangent-radius theorem the radius drawn to the point of contact meets the tangent at 90 degrees, so the centre of the table is exactly 60 cm from the wall.