Ex 11.2, 2
Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation.
Tangents on the given circle can be drawn as follows.
- Draw a circle of 4 cm radius with centre as O on the given plane.
- Draw a circle of 6 cm radius taking O as its centre.
- Locate a point P on this circle and join OP.
- Bisect OP. Let M be the mid-point of OP
- Take M as its centre and MO as its radius, draw a circle. Let it intersect the given circle at the points Q and R.
- Join PQ and PR.
PQ and PR are the required tangents.
Lengths of PQ and PR is 4.47 m
Finding lengths of PQ and PR
Join OQ and OR
Since tangent is perpendicular to radius
∠ PQO = 90° and ∠ PRO = 90°
Thus, Δ PQO is a right angled triangle,
PO = radius of bigger circle = 6 cm
and OQ = radius of smaller circle = 4 cm
By Pythagoras theorem
PO2 = PQ2 + OQ2
62 = PQ2 + 42
36 = PQ2 + 16
PQ2 = 36 – 16
PQ2 = 20
PQ = √20 = √(5 ×4) = √4 × √5 = 2√5
PQ = 2 × 2.236
PQ = 4.47 cm
Similarly, PR = 4.47 cm
We need to prove that PQ and PR are the tangents to the circle.
Join OQ and OR.
∠PQO is an angle in the semi-circle
of the blue circle
And we know that angle in a
semi-circle is a right angle.
∴ ∠PQO = 90°
⇒ OQ ⊥ PQ
Since OQ is the radius of the circle,
PQ has to be a tangent of the circle.
Similarly, PR is a tangent of the circle.