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Two Tangents from a Point (Cambridge (CIE) IGCSE International Maths): Revision Note

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on

Two Tangents from a Point

Circle Theorem: Tangents from an external point are equal in length

  • Two tangents from the same external point are equal in length

  • This means that a kite can be formed by two tangents meeting a circle

    • The kite below has a vertical line of symmetry

      • It is formed from two congruent triangles back-to-back

    • The kite will have two right angles where the tangents meet the radii

      • You can use Pythagoras and SOHCAHTOA on each of these triangles

A circle with centre, O, and two points on the circumference, R and S. Tangents to the circle at these two points, intersect at a point outside the circle, T. OR and OS are radii. ROST forms a kite.

Examiner Tips and Tricks

  • Look for tangents in the exam and draw on the radius at right angles to see if it helps

Worked Example

Consider the diagram below.

A circle with centre, O, and two points on the circumference, S and R. Tangents to the circle, at S and R, meet at a point outside the circle, T. OR = 3 cm, OT = 11 cm.

(a) Find the length ST.

The lines ST and RT are both tangents to the circle
They meet the two radii on the circumference at the points S and T

Angle TSO = 90° because a radius and a tangent meet at right angles

Use Pythagoras' theorem on the right-angled triangle OST to calculate the length ST

table row cell 3 squared plus open parentheses S T close parentheses squared end cell equals cell 11 squared end cell row cell 9 plus open parentheses S T close parentheses squared end cell equals 121 row cell open parentheses S T close parentheses squared end cell equals 112 row cell S T end cell equals cell square root of 112 equals 10.58300... end cell end table

ST = 10.6 cm (3 s.f.)

(b) Find the length RT.

Two tangents from the same external point are equal in length

ST = RT

RT = 10.6 cm (3 s.f.)

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    Author
    Reviewer
    Naomi C

    Author: Naomi C

    Expertise: Maths

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.