Question 5: Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

**Answer:** For this, draw a circle with centre O and tangent AB which touches the circle at point P.

Let us assume a point O’ which does not coincide with centre O.

Let us assume that PO’ is perpendicular to AB.

But PO will be perpendicular to AB because radius is always perpendicular to tangent.

Additionally, the figure shows that ∠O’PB < ∠OPB

According to our assumption, these angles should have been right angles.

But since we assumed that O and O’ are not coincident, hence both of them cannot be right angles.

Hence, it is proved that the perpendicular at the point of contact to the tangent to a circle passes through centre.

Question: 6 - The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

**Answer:** Here; OA = 5 cm, AB = 4 cm, OB = ?

Using Pythagoras Theorem,

In ΔOBA

`OB^2 = OA^2 – BA^2`

`= 5^2 – 4^2`

`= 25 – 16 = 9`

Or, `OB = 3` cm

Question: 7 - Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

**Answer:** Here; OA = 5 cm, OP = 3 cm, AB = ?

In ΔOPA,

`AP^2 = OA^2 – OP^2`

`= 5^2 – 3^2`

`= 25 – 9 = 16`

Or, `AP = 4` cm

Since OP bisects AB

Hence, `AP = AB`

So, `AB = 2 xx 4 = 8` cm

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