Tangent to a Circle (Concept) MCQs Quiz | Class 10

This quiz covers essential concepts on Tangent to a Circle, including the definition of a tangent and the point of contact, for Class X Mathematics (Code 041), Unit IV: Geometry. Test your understanding, then submit to see your score and download a detailed answer PDF.

Understanding Tangents to a Circle

A circle is a fundamental geometric shape, and understanding the lines related to it is crucial. Among these, tangents hold a special significance due to their unique properties and applications in various mathematical problems. This section aims to reinforce your understanding of tangents and related concepts.

1. Tangent to a Circle: Definition

A tangent to a circle is a straight line that touches the circle at exactly one point. This means that the tangent line does not cross into the interior of the circle; it only makes contact at a single boundary point. Think of a ruler placed gently along the edge of a coin—it touches at just one spot.

  • A line that intersects a circle at two distinct points is called a secant.
  • A line that does not intersect a circle at all is just a non-intersecting line.
  • There can be infinitely many tangents to a circle, each touching at a different point on its circumference.

2. Point of Contact

The single point where a tangent line touches the circle is called the point of contact (or point of tangency). This point is unique for any given tangent.

Key Property (Theorem 1): The tangent at any point of a circle is perpendicular to the radius through the point of contact. This is a very important theorem. It implies that if you draw a line from the center of the circle to the point of contact, this line (which is a radius) will form a 90-degree angle with the tangent line. This property is extensively used in solving problems involving tangents.

3. Number of Tangents from a Point

  • Case 1: Point inside the circle: No tangent can be drawn to a circle from a point lying inside it. Any line passing through an interior point will always intersect the circle at two points (a secant).
  • Case 2: Point on the circle: There is exactly one tangent to a circle passing through a point lying on the circle. The radius to this point of contact is perpendicular to the tangent.
  • Case 3: Point outside the circle: From an external point, exactly two tangents can be drawn to a circle.

Key Property (Theorem 2): The lengths of tangents drawn from an external point to a circle are equal. If P is an external point and PA and PB are two tangents to a circle with center O, then PA = PB. Also, the line segment PO bisects the angle APB and angle AOB.

Quick Revision Summary

  • A tangent touches a circle at exactly one point.
  • The point where the tangent touches the circle is the point of contact.
  • Radius at the point of contact is perpendicular to the tangent.
  • From an external point, two tangents can be drawn to a circle.
  • Lengths of tangents from an external point to a circle are equal.

Practice Questions

  1. If a circle has a radius of 5 cm and a tangent is drawn at point A, what is the angle between the radius OA and the tangent at A?
  2. How many tangents can be drawn to a circle from a point lying on the circle?
  3. A point P is 13 cm away from the center of a circle. The radius of the circle is 5 cm. Find the length of the tangent drawn from P to the circle.
  4. If two tangents PA and PB are drawn from an external point P to a circle with center O, and angle APB = 60 degrees, what is the measure of angle AOB?
  5. What is the maximum number of parallel tangents a circle can have?

(Answers: 1. 90 degrees; 2. One; 3. 12 cm (using Pythagoras theorem); 4. 120 degrees; 5. Two)