Tangents to Circles

≈ 36 min

Tangents to Circles

A tangent touches a circle at exactly one point. Everything about tangent construction follows from one property: a tangent is perpendicular to the radius at the point of contact.

Tangent at a given point on the circle. Draw the radius to that point and erect a perpendicular there. Done.

Tangent from a point outside the circle. You do not know the contact point yet, so you must find it. Join the external point P to the centre O. Bisect OP to find its midpoint, and draw a circle on OP as diameter. Where that circle cuts the given circle is the contact point — because any angle in a semicircle is 90°, guaranteeing the radius meets the tangent at a right angle. There are always two such tangents.

Worked example. A belt running around two pulleys is a pair of common tangents. Drawing the belt by eye produces a line that visibly cuts into or floats off the pulley; the construction puts it exactly on the contact point.

Core checkpoint: Find the point of contact first. Never draw a tangent by sliding a ruler until it looks right.

At the point of contact, what angle in degrees does a tangent make with the radius drawn to that point?

Why does drawing a circle on OP as diameter locate the point of contact?

Any angle in a semicircle is 90°, so where it cuts the given circle the radius meets the tangent at a right angle.

Because that circle is the same size as the given circle.

Because it passes through the centre, which is always the contact point.

Because it halves the distance from P to the circle.

How many tangents can be drawn to a circle from a single point outside it?

A learner draws a tangent from P to the point on the circle that looks nearest to P. Why is this wrong?

The nearest point lies on the line OP, where a tangent would be perpendicular to OP and could not pass through P at all.

It is correct — the nearest point is always the contact point.

It is wrong only if the circle is very small.

The nearest point cannot be found without a ruler.