Two Tangents from a Point (Cambridge International IGCSE Maths)

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

Exam Hint

• 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) 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*

$\begin{gathered} 3^{2} + ( S T )^{2} & = 11^{2} \\ 9 + ( S T )^{2} & = 121 \\ ( S T )^{2} & = 112 \\ S T & = \sqrt{112} = 10.58300... \end{gathered}$

Final answer: equationST = 10.6 cm (3 s.f.)**

(b) Find the length RT.

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

ST = RT

Final answer: RT = 10.6 cm (3 s.f.)**