Tangent & Radius (Cambridge International IGCSE Mathematics)

Tangent & Radius

What is a tangent?

• A tangent to a circle is a straight line outside of the circle that touches its circumference only once

Circle Theorem: A radius and a tangent are perpendicular

• This circle theorem states that a radius and a tangent meet at right angles (90°)

  • This may also be described as being perpendicular to each other

• When using this theorem in an exam you must use the keywords

  • A radius and a tangent meet at right angles(or 90°)

Exam Hint

• If you spot a tangent on a circle diagram, look to see if it meets a radius and label the right angle on the diagram

  • In some cases just doing this can earn you a mark!

• If you think you have spotted this circle theorem in a question, make sure it is a radius that meets the tangent, and not a chord

  • A radius passes through the centre of the circle

Worked Example P and Q are points on the circle, centre O.

APB is a tangent to the circle at P.

(i) Explain why angle OPB is 90°.

(ii) Find the value of $x.$

Answer:

(i)**

Final answer: Angle OPB is 90° because the angle between a tangent and a radius is 90° (and OP is a radius, and APB is a tangent).

(ii) As angle* OPB **is 90°, we can find angle **OPQ *

OPQ = 90 - 53 = 37° *

As OP and OQ are both the radius of the circle, they have the same length. This means the triangle* OPQ** is isosceles, so the base angles (OPQ and OQP) are equal *

Using the fact that the internal angles in a triangle sum to 180, we can find angle $x$

$\begin{gathered} x + 37 + 37 & = 180 \\ x + 74 & = 180 \\ x = 180 - 74 \end{gathered}$

Final answer: $x=106$