Constructing Tangents to a Circle

Definition: A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of contact or point of tangency.

Key properties used in construction:

• The tangent at any point is perpendicular to the radius through that point.
• From an external point, exactly two tangents can be drawn to a circle.
• These two tangents are equal in length.
• The angle in a semicircle is 90° — this property helps locate the points of tangency.

Justification — Construction from External Point:

Given circle with centre O and radius r, and external point P at distance d from O.

1. Join OP.
2. Find midpoint M of OP (by perpendicular bisector construction).
3. With M as centre and MO as radius, draw a circle (this circle has diameter OP).
4. This new circle intersects the given circle at points T₁ and T₂.
5. Why T₁ and T₂ are points of tangency:
6. T₁ lies on the circle with diameter OP, so ∠OT₁P = 90° (angle in semicircle).
7. This means OT₁ ⊥ PT₁, i.e., the radius OT₁ is perpendicular to PT₁.
8. Since PT₁ is perpendicular to the radius at the point on the circle, PT₁ is a tangent.
9. Similarly, PT₂ is a tangent.
10. Both tangents have equal length: PT₁ = PT₂ = √(d² − r²).

Why two tangent points exist:

• The auxiliary circle (diameter OP) has radius OP/2, which is greater than r (since P is external, OP > r, so OP/2 > r/2).
• The auxiliary circle passes through O (inside the original circle) and P (outside the original circle).
• Since it crosses from inside to outside, it must intersect the original circle at exactly two points.
• These two intersection points are the tangent points T₁ and T₂.

Type 1: Tangent from a Point ON the Circle

• The point of tangency is already known.
• Construct the radius to that point, then draw a perpendicular to the radius at that point.
• The perpendicular line is the tangent.
• There is only one tangent at a given point on the circle.

Type 2: Tangent from an External Point

• The external point P is at a distance greater than the radius from the centre.
• Construct the perpendicular bisector of OP to find midpoint M.
• Draw a circle with centre M and radius MO.
• This circle intersects the original circle at two points — the points of tangency.
• Two tangents are drawn, and they are equal in length.

Number of tangents based on position of point:

• Point inside the circle (d < r): Zero tangents. Every line through an internal point is a secant.
• Point on the circle (d = r): Exactly one tangent, perpendicular to the radius.
• Point outside the circle (d > r): Exactly two tangents, equal in length.

Angle between two tangents (from external point):

• sin(half-angle) = r/d, so the full angle between tangents = 2 arcsin(r/d).
• When d is very large compared to r, the angle approaches 0 degrees (tangents become nearly parallel).
• When d is just slightly greater than r, the angle approaches 180 degrees (tangents open wide).

Construction A — Tangent from a Point on the Circle:

1. Draw a circle with centre O and given radius.
2. Mark the given point T on the circle.
3. Join OT (this is the radius).
4. At point T, construct a perpendicular to OT using compass (standard perpendicular construction at a point on a line).
5. The perpendicular line through T is the required tangent.

Construction B — Two Tangents from an External Point:

1. Draw a circle with centre O and radius r.
2. Mark the external point P at distance d from O (d > r).
3. Join OP.
4. Find the midpoint M of OP using perpendicular bisector construction.
5. With M as centre and MO (= MP = OP/2) as radius, draw a circle.
6. This circle intersects the original circle at T₁ and T₂.
7. Join PT₁ and PT₂.
8. PT₁ and PT₂ are the two required tangents.

How to find perpendicular bisector (Step 4):

1. With O as centre and radius more than half of OP, draw arcs on both sides.
2. With P as centre and the same radius, draw arcs intersecting the previous ones.
3. Join the two intersection points — this line bisects OP at M.

Verification Steps:

1. After construction, measure PT₁ and PT₂ — they should be equal.
2. Measure ∠OT₁P with a protractor — it should be 90°.
3. Check: PT₁² + OT₁² should equal OP² (Pythagoras).

Common Mistakes:

• Perpendicular bisector error: If M is not exactly the midpoint of OP, the auxiliary circle will not pass through the correct tangent points.
• Wrong compass radius for auxiliary circle: The radius must be MO (= MP = OP/2), not OP or r.
• Not drawing both tangent lines: The auxiliary circle intersects the original circle at two points — both must be joined to P.
• Confusing radius and diameter: The auxiliary circle has OP as diameter (not radius).

Special Constructions:

• Direct common tangent: For two circles, tangents that touch both circles on the same side.
• Transverse common tangent: Tangents that cross between the two circles. These are advanced constructions beyond Class 10 scope.

Engineering — Gear Design:

• Tangent lines to circular gears determine the contact line between meshing gears.
• Belt drives wrap around pulleys along tangent lines.

Optics — Light Rays:

• A light ray tangent to a curved mirror or lens surface determines the angle of reflection/refraction.

Navigation:

• A ship sailing past a circular island follows a tangent path to avoid the island.
• The closest approach distance equals the tangent length calculation.

Road Design:

• A straight road meeting a circular roundabout enters along a tangent to the roundabout.

Astronomy:

• The line of sight from Earth to the edge of a planet is a tangent to the planet's circular cross-section.