The tangent and the radius
A tangent is a straight line that touches a circle at exactly one point, the point of contact, and does not cut into it. The radius drawn to the point of contact meets the tangent at a right angle.
The reason is that every other point of the tangent lies outside the circle, so the point of contact is the point of the tangent nearest the center, and the shortest line from a point to a straight line meets it at a right angle.
The tangent at A, and the radius from the center to A. They meet at a right angle.
Two tangents from one point
Take a point P outside the circle. Exactly two tangents can be drawn from P: one touching the circle at A, and one touching it at B. Each tangent meets its own radius at a right angle: if O is the center, the angles OAP and OBP are both 90°.
The lengths PA and PB, from P along each tangent to its point of contact, are called the lengths of the tangents from P.
Two tangents from P, touching the circle at A and at B, one above the other. Each meets the radius from the center at a right angle.
The two tangents are equal
Join P to the center O. This makes two triangles, OAP and OBP. Compare them.
Both have a right angle, at A and at B. Their hypotenuses are the same line, OP. And OA = OB, because both are radii. A right angle, the hypotenuse and one other side: the two triangles are congruent by RHS.
Congruent triangles have all their parts equal, so the third sides are equal: PA = PB. The two tangents from a point outside a circle are the same length.
The dashed line joins P to the center. It is the hypotenuse of both right triangles, the radii are equal, and so the two tangents, each marked with a tick, are equal.
The line to the center cuts the angles in half
The congruent triangles also have equal angles. The angle OPA equals the angle OPB, so the line PO cuts the angle APB between the two tangents exactly in half. And the angle AOP equals the angle BOP, so it cuts the angle AOB at the center in half too.
The quadrilateral OAPB has angles of 90° at A and at B. Its four angles add to 360°, so the angle at P and the angle at O add to 360° − 90° − 90° = 180°. If the tangents meet at 50°, the angle AOB is 180° − 50° = 130°. Then OPA = 50° ÷ 2 = 25° and AOP = 130° ÷ 2 = 65°. Check triangle OAP: 90° + 25° + 65° = 180°.
Finding a length with Pythagoras
Each triangle OAP is right-angled at A, with OP as its hypotenuse, so Pythagoras' theorem connects the radius, the distance to the center and the length of the tangent: PA² = OP² − OA².
A circle has a radius of 6 cm, and P is 10 cm from its center. Then PA² , so PA cm, and PB = 8 cm as well. Check: .
The radius is 6 and the dashed line from P to the center is 10, so each tangent is .
The usual mistakes
Adding the squares. The right angle is at A, where the tangent meets the radius, so OP is the hypotenuse, and the tangent is found by subtracting: , not .
An angle other than 90° between the tangent and the radius. A line that met the radius at 45° would cut into the circle and cross it twice, so it would not be a tangent.
Doubling the length for the second tangent. The two tangents from one point are equal: if PA is 8 cm, PB is 8 cm, not 16 cm.
Worked example: Two Paths from a Lamp Post That Touch a Round Pond
Question A circular pond has center O and radius 8 m. A lamp post stands at a point P that is 17 m from O. Two straight paths run from P and each just touches the edge of the pond, one at A and the other at B. (a) How long is the path PA? (b) A low fence runs from A to P and on from P to B. How long is the fence?
1.The path PA just touches the pond at A, so PA is a tangent and the radius OA is perpendicular to it: angle OAP is 90°.
PA just touches the pond, so it is a tangent and the radius OA is at right angles to it. 2.Triangle OAP is right-angled at A, with hypotenuse OP = 17 m and OA = 8 m. By Pythagoras' theorem, PA2 = 172 − 82 = 289 − 64 = 225.
Triangle OAP is right-angled at A with hypotenuse OP = 17: PA2 = 172 − 82 = 225. 3.(a) PA = √225 = 15 m.
(a) PA = √225 = 15 m. 4.PA and PB are the two tangents from the same point P, so they are equal: PB = PA = 15 m. In the same way, triangle OBP is right-angled at B with OP = 17 m and OB = 8 m, which gives PB = 15 m directly.
The two tangents from P are equal, so PB = PA = 15 m. 5.(b) The fence is PA + PB = 15 + 15 = 30 m long. Check: 82 + 152 = 64 + 225 = 289 = 172.
(b) The fence is 15 + 15 = 30 m long.
Answer: (a) 15 m; (b) 30 m
Common mistakes
- Taking OP as a shorter side and adding the squares: 172 + 82 = 353. The right angle is at A, where the tangent meets the radius, so OP is the hypotenuse and the squares must be subtracted.
- Measuring the fence as 15 + 8 m by running it through the center. The fence runs along the two tangents, from A to P and from P to B, and both are 15 m.
More congruence, similarity and circle theorems problems, worked step by step →