Home > Geometry > Circle Theorems 2

Circle Theorems 2

Tangents to circles

Investigation

Theorem 1

The length of the tangents from a point to a circle are of equal length

Theorem 2

The Angle between a tangent and a radius is a right-angle

Theorem 3

The angel between a tangent and a cord is equal to the angle subtended by the cord

Examples

Given that PA is a tangent find the value of x
Since PA is a tangent \angle PAO = 90^o
\implies x = 180^o-90^o-65^o=25^o \because 180^o \mbox{in a }\triangle

Given that PA and PB are tangents find the value of x
Since PA and PB are tangents they must both be of equal length.  Therefore \triangle ABP is isosceles.
\implies \angle ABP =\frac{180^o-49^o}{2}=65.5^o \because base angles in an isosceles triangle are equal.
\implies x=90^o-65.5^o=24.5^o \because the angle between a tangent and a radius is a right-angle.

Find the value of x
Since \triangle ABC \mbox{ is isosceles}\implies \angle ACB = 50^o
\implies x=50^o \because \mbox{Angle subtended by the cord}

Questions

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Categories: Geometry
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