What is a tangent?
A tangent is a straight line that touches a circle at exactly one point, called the point of contact. It never crosses into the circle.
Key idea
A tangent is perpendicular to the radius at the point of contact (angle = 90°). Two tangents drawn from the same external point are equal in length.
Right angle at the contact
Because the radius meets the tangent at 90°, any triangle formed by a radius, a tangent and the line to the centre is right-angled. If OA is a radius (5 cm) and PA is a tangent (12 cm), then OP = √(5² + 12²) = √169 = 13 cm by Pythagoras.
Worked example
From external point P, tangents PA and PB touch a circle, centre O. Angle APB = 50°. In quadrilateral OAPB the angles at A and B are 90° each, so angle AOB = 360 − 90 − 90 − 50 = 130°.
Equal tangents
Since PA = PB, triangle PAB is isosceles, and the line PO bisects angle APB. So each half of angle APB here is 50 ÷ 2 = 25°. The same line PO also bisects angle AOB at the centre, which is a quick way to split a symmetric tangent figure into two equal right-angled triangles.
Because every tangent gives a right angle with its radius, these problems almost always turn into Pythagoras or angle-sum work. Spot the radius drawn to the point of contact first, mark the 90°, and the rest of the figure usually falls into place.
Remember
- Tangent ⊥ radius at the point of contact.
- Two tangents from one point are equal in length.
- The line from the point to the centre bisects the angle between the tangents.