# Draw a circle with the help of a bangle. Take a point outside the circle. Construct the pair of tangents from this point to the circle.

**Solution:**

**Steps of construction:**

Draw any circle using a bangle.

To find its centre:

- Draw two chords on the circle say AB and CD.
- Draw the perpendicular bisectors of AB and CD to intersect at O.

Now, ‘O’ is the centre of the circle (since the perpendiculars drawn from the centre of a circle to any chord bisect the chord and vice versa).

To draw the tangents from a point ‘P’ outside the circle:

- Take a point P outside the circle and draw the perpendicular bisector of OP which meets at OP at O’.
- With O’ as the centre and OO’ as radius draw a circle that cuts the given circle at Q and R.
- Join PQ and PR.

Thus, PQ and PR are the required tangents.

Proof:

∠QOP = ∠ORP = 90° (Angle in a semi-circle)

∴ OQ ⊥ QP and OR ⊥ RP. (We know that the line joining the centre of a circle to the tangent is always perpendicular)

Hence, we have PQ and PR as the tangents to the given circle.

**☛ Check: **NCERT Solutions Class 10 Maths Chapter 11

**Video Solution:**

## Draw a circle with the help of a bangle. Take a point outside the circle. Construct the pair of tangents from this point to the circle.

NCERT Solutions Class 10 Maths Chapter 11 Exercise 11.2 Question 7

**Summary:**

A circle was drawn with the help of a bangle. PQ and PR are the required tangents constructed to the given circle from a point P outside the circle.

**☛ Related Questions:**

- Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameters each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q.
- Draw a pair of tangents to a circle of radius 5 cm which is inclined to each other at an angle of 60°.
- Draw a line segment AB of length 8 cm. Taking A as the centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct a tangent to each circle from the centre of the other circle.
- Let ABC be a right triangle in which AB = 6 cm, BC = 8 cm and ∠B = 90°. BD is perpendicular to AC. The circle through B, C and D is drawn. Construct the tangents from A to this circle.

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