In each of the following, give also the justification of the construction:
Draw a circle of radius 6 cm. From a point 10 cm away from its center, construct the pair of tangents to the circle and measure their lengths.

Steps of construction:

1. Take a point O as the centre and 6 cm radius. Draw a circle.
2. Take a point P in such a manner that OP = 10cm
3. With O and P as centres and radius more than half of OP draw arcs above and below OP to intersect at X and Y.
4. Draw line segment XY to intersect OP at M.
5. With M as the centre and OM as radius draw a circle to intersect the given circle at Q and R.
6. Join PQ and PR. PQ and PR are the tangents, where PQ and PR = 8cm.

Proof:

Since PQ ⊥ OQ (Angle in a semicircle)

∠PQO = 90º

In right ΔPQO,

OP = 10 cm, OQ = 6cm (radius)

PQ² = OP² - OQ²

= (10)² - (6)²

= 100 - 36

= 64

PQ = √64

= 8 cm

Similarly, we can prove that PR = 8 cm.