Introduction

In Class 10 mathematics under the West Bengal Board, Chapter 17—Construction: Construction of Tangent to a Circle—focuses on fundamental geometric constructions. This section helps students understand and draw tangents using classical ruler‑and‑compass methods. Mastery of this topic strengthens understanding of circle geometry: properties of tangents, perpendicular radii, and angular relationships. Key topic keywords include: tangent to a circle, radius perpendicular to tangent, construction with compass, angle between radius and tangent, point outside circle, point on circle.

1. Definition and Key Principle

A tangent to a circle is a straight line that touches the circle at exactly one point. The defining property: the radius drawn to the point of contact is perpendicular to the tangent. This is the core geometric principle used in every construction under this chapter.

2. Types of Constructions in the Syllabus

The syllabus includes two classical construction problems:

A. Constructing a tangent when the point lies on the circle

Given a circle with center O and a point P on the circumference, the tangent at point P is the line perpendicular to OP at P.

Steps:

1. Draw radius OP.

2. At P, construct a perpendicular line to OP using the standard ruler‑and‑compass method:
   • With P as centre, draw arcs on both sides of OP.

   • From those arc‑intersections, draw arcs that intersect above and below OP.

   • Join the intersection points to get the perpendicular through P.

Example:
Circle with centre O, point P on the circle. Draw OP. Using compass, construct a perpendicular at P. That line is the tangent at P.

Keywords: perpendicular at point on circle, tangent at point on circumference, radius perpendicular to tangent.

B. Constructing tangents from a point outside the circle

Given circle with centre O and radius r, and a point P outside the circle, construct two tangents from P to the circle. This is a common construction and requires the right‑angle property and the concept of circle of Apollonius.

Steps:

1. Join P to O; draw segment PO.

2. Construct a circle with OP as diameter.

3. Let this new circle meet the given circle at points A and B.

4. Join P to A and P to B. PA and PB are the required tangents.

Example:
Circle centre O, radius 4 cm. External point P such that OP = 6 cm. Construct circle with OP as diameter, find intersection points A, B with the given circle, then draw PA and PB. These are the tangents.

Keywords: tangents from external point, circle with diameter OP, two tangents to circle, construction from external point.

3. Underlying Geometry and Justification

• Perpendicularity: Radius OP ⟂ tangent at P (on the circle).

• Right‑angle in semicircle: Angle subtended by a diameter is a right angle. In construction B, angle OAP = 90° and OBP = 90°, meaning PA and PB are perpendicular to radii and thus tangents.

These justifications affirm that the lines constructed are indeed tangents.

4. Step‑by‑Step with Emphasis on Accuracy and Clarity

A. Tangent at a point on the circle

1. Draw radius OP accurately.

2. Use the compass to mark equal arcs across OP.

3. Intersect the arcs to get two points.

4. Join them to create the perpendicular line through P.

B. Tangents from an external point

1. Accurately measure OP.

2. Construct the auxiliary circle with OP as diameter.

3. Find precise intersection points A and B.

4. Join P to A and P to B.

Clear labeling of points O, P, A, B is essential for clarity.

5. Examples with Numerical Values

• Example 1: Circle with centre O and radius 5 cm; P on the circle. Construct tangent at P.
  • Result: Line through P perpendicular to OP.

• Example 2: Circle centre O, radius 3 cm; external point P such that OP = 7 cm. Construct the two tangents.

Construct circle with diameter OP = 7 cm, find its intersections A, B, then draw PA and PB.