Circles
Circles and tangents
Use the two tangent properties of a circle: a tangent meets the radius at a right angle, and the two tangents from an outside point are equal.
Slide a straight line towards a circle. While it is well clear, it misses. Push it further and it cuts across, meeting the circle at two points. There is exactly one position in between where it meets the circle at a single point and nowhere else — that line is a tangent, and the point is where it touches.
Property one
The radius drawn to the point of contact is perpendicular to the tangent. Every tangent question starts by drawing that radius in, because it is what turns the picture into a right-angled triangle you already know how to handle.
Worked example
A circle has radius 5 and a point P is 13 from the center. How long is the tangent from P?
Draw the radius OA to the point of contact A. Angle OAP is 90°.
Without this the picture has no right angle in it and Pythagoras has nothing to bite on.
Property two
The two tangents drawn from the same outside point are equal in length, and the line from that point to the center bisects the angle between them. Both follow from the two right-angled triangles being congruent: equal radii, a shared hypotenuse, right angles at the points of contact.
Have a play
Three straight lines have been drawn on this circle. Find out what each one is called.
Tap any part of the picture.
Try it together
Two tangents from a point P touch a circle with center O at A and B, and the angle APB is 70°. Let us find the angle AOB.
Join OA, OB and OP. Now OAPB is a quadrilateral with right angles at A and B.
1.What do the four angles of any quadrilateral add up to, in degrees?
Have a go
Have a go on your own: two tangents from a point P touch a circle with center O at A and B, and the angle APB is 60°. What is the angle AOB, in degrees?
Hint: The angle at O and the angle at P always add to 180°.
Ready to practice?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.