Name: Construction of Tangents & Perpendicular Bisector
Uploaded: 2019-07-19
Duration: 3 min 50 s
Description: We can construct two tangents from an external point to a circle. In this video, we will learn how to construct tangents to a circle from a point outside a circle.

Question

$PA$ and $PB$ are tangents drawn to a circle of centre $O$ from an external point $P$ . Chord $AB$ makes an angle of $30 °$ with the radius at the point of contact.

If length of the chord is $6 cm$ , find the length of the tangent $PA$ and the length of the radius $OA$ .

Accepted Answer

Finding $PA$ ,

Given $∠ OAB = 30^{°}$

We know that,

Tangent at any point the circle is perpendicular to radius at the point of contact.

So, $OA ⊥ AP$

$∴ ∠ OAP = 90^{°}$

Thus, $∠ OAB + ∠ PAB = 90^{°}$

$30^{°} + ∠ PAB = 90^{°}$

$∠ PAB = 60^{°}$

Also, since

Tangents from external point are equal.

$PA = PB$

And we know angles opposite to equal sides are equal.

Thus, $∠ PAB = ∠ PBA = 60^{°}$

In $△ APB$ , By angle sum property

$∠ APB + ∠ PAB + ∠ PBA = 180^{°}$

$∠ APB + 60^{°} + 60^{°} = 180^{°}$

$∠ APB + 120^{°} = 180^{°}$

$∠ APB = 60^{°}$

Thus, in $△ APB$ , $∠ PAB = ∠ PBA = ∠ APB = 60^{°}$

Since, all angles are $60^{°}$ .

$∴ △ APB$ is an equilateral triangle and in equilateral triangle, all sides are equal.

$∴ PA = AB$

$PA = 6 cm$

Finding radius OA

Since $∠ APB = 60^{°}$

By symmetry

$∠ OPA = \frac{∠ APB}{2} = \frac{60^{°}}{2} = 30^{°}$

Using Trigonometry in right $△ AOP$

tan P $= \frac{Side opposite to angle P}{Side adjacent to angle P}$

tan $30^{°} = \frac{OA}{PA}$

$\frac{1}{\sqrt{3}} = \frac{OA}{6}$

OA $= \frac{6}{\sqrt{3}}$

OA $= \frac{2 \times 3}{\sqrt{3}}$

OA $= 2 \sqrt{3}$ cm

Hence, the length of the tangent $PA$ is $6 cm$ and the length of the radius $OA$ is $2 \sqrt{3} cm$ .

$PA$ and $PB$ are tangents drawn to a circle of centre $O$ from an external point $P$ . Chord $AB$ makes an angle of $30 °$ with the radius at the point of contact.

If length of the chord is $6 cm$ , find the length of the tangent $PA$ and the length of the radius $OA$ .

Important Questions on Circles

Learn and Prepare for any exam you want !

PA and PB are tangents drawn to a circle of centre O from an external point P. Chord AB makes an angle of 30° with the radius at the point of contact. If length of the chord is 6 cm, find the length of the tangent PA and the length of the radius OA.

Important Questions on Circles

Learn and Prepare for any exam you want !

$PA$ and $PB$ are tangents drawn to a circle of centre $O$ from an external point $P$ . Chord $AB$ makes an angle of $30 °$ with the radius at the point of contact.

If length of the chord is $6 cm$ , find the length of the tangent $PA$ and the length of the radius $OA$ .