Construction of Tangent to a circle
Multiple Choice Questions (MCQ)
1. What is the defining property of a tangent to a circle?
A) It cuts the circle at two points
B) It is parallel to the radius
C) It touches the circle at exactly one point
D) It passes through the center of the circle
Answer: C) It touches the circle at exactly one point
2. What is the angle between the radius and the tangent at the point of contact?
A) 30°
B) 60°
C) 90°
D) 180°
Answer: C) 90°
3. In constructing a tangent from a point on the circle, what must be drawn first?
A) A line parallel to the radius
B) A diameter
C) The radius to the point
D) The center of the circle
Answer: C) The radius to the point
4. In the construction of two tangents from a point outside the circle, what is constructed using OP as diameter?
A) A triangle
B) A semicircle
C) A suare
D) A circle
Answer: D) A circle
5. Why is the angle ∠OAP = 90° when OP is a diameter?
A) It is a property of triangles
B) It is assumed
C) Angle in a semicircle is 90°
D) O is the midpoint
Answer: C) Angle in a semicircle is 90°
Short Answer Questions (SAQ)
1. Define a tangent to a circle.
Answer: A tangent to a circle is a straight line that touches the circle at exactly one point and is perpendicular to the radius at that point.
2. What is the first step when constructing tangents from a point outside a circle?
Answer: The first step is to join the external point P to the center O, creating the segment OP.
3. Why do we construct a circle with OP as diameter while drawing tangents from an external point?
Answer: Because the angle in a semicircle is 90°, and this helps to locate the exact points where the tangents touch the circle.
Long Answer Questions (LAQ)
1. Construct a tangent to a circle from a point P on the circle with center O. Explain the steps.
Answer: Steps:
Draw the radius OP.
Using a compass, draw two arcs on both sides of OP from point P.
From the intersections of those arcs, draw two new arcs above and below OP to intersect.
Join these intersections to form a line perpendicular to OP at point P.
This line is the tangent at point P.
Justification: The radius OP is perpendicular to the tangent at P.
2. Construct two tangents from an external point P to a circle with center O and radius 3 cm, where OP = 7 cm. Describe the construction and reasoning.
Answer: Steps:
Draw the circle with center O and radius 3 cm.
Mark an external point P such that OP = 7 cm.
Join O and P to form segment OP.
Construct a circle with OP as the diameter.
Let this auxiliary circle intersect the original circle at points A and B.
Join PA and PB. These are the reuired tangents.
Justification: Since ∠OAP = ∠OBP = 90° (angle in a semicircle), PA and PB are perpendicular to the radii and are tangents to the circle.