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.
Steps of construction
Draw a circle of radius 4 cm with center O
Draw concentric circle of radius 6 cm, with center O
Mark point P on the larger circle
3. Join PO.
Make perpendicular bisector of PO
Let M be the midpoint of PO.
4. Taking M as centre and MO as radius,
draw a circle.
5. Let it intersect the given circle at points Q and R.
6. Join PQ and PR.
∴ PQ and PR are the required two tangents.
After measuring, lengths of tangents PQ and PR are 4.47 cm each
Finding lengths of PQ and PR
Join OQ and OR
Since tangent is perpendicular to radius
∠PQO = 90° & ∠PRO = 90°
Thus,
Δ PQO is a right angled triangle,
And,
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
PQ = 2 × 2.236
PQ = 4.47 cm
Similarly,
PR = 4.47 cm
Justification
We need to prove that PQ and PR are the tangents to the circle.
Join OQ and OR.
Now,
∠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.

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.