Question 3 - Constructing tangents to a circle - Constructions

Last updated at May 26, 2023 by Teachoo

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Question 3
Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q.
Steps of construction
Draw a circle of radius 3 cm
Draw diameter of circle, and extend it
and mark points P and Q, 7 cm from the center
Let’s first draw tangent from point P
3. 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 A and B.
6. Join PA and PB.
Now, we draw tangent from point Q
Similarly we draw tangent from point Q
∴ QC and QD are the tangents from point Q
Justification
We need to prove that PA, PB, QC, QD are the tangents to the circle.
Join OA, OB, OC and OD
Now,
∠PAO is an angle in the semi-circle of the blue circle
And we know that,
Angle in a semi-circle is a right angle.
∴ ∠PAO = 90°
⇒ OA ⊥ PA
Since OA is the radius of the circle,
PA has to be a tangent of the circle.
Similarly, we can prove
PB, QC, QD are tangents of the circle.

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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