CBSE Class 10-Mathematics: Chapter –11 Constructions Part 15
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Question 26:
Draw a circle of radius . Take two points P and Q on one of its extended diameter each at a distance of from its centre. Draw tangents to the circle from these two points P and Q.
Answer:
To construct: A circle of radius and take two points P and Q on one of its extended diameter each at a distance of from its centre and then draw tangents to the circle from these two points P and Q.
Steps of Construction:
Bisect PO. Let be the mid-point of PO.
(b) Taking M as center and MO as radius, draw a circle. Let it intersects the given circle at the points A and B.
(c) Join PA and PB.
Then PA and PB are the required two tangents.
(d) Bisect QO. Let N be the mid-point of QO.
(e) Taking N as center and NO as radius, draw a circle. Let it intersects the given circle at the points C and D.
(f) Join QC and QD.
Then QC and QD are the required two tangents.
A Circle of Radius
Justification: Join OA and OB
Then is an angle in the semicircle and therefore
Since OA is a radius of the given circle, PA has to be a tangent to the circle. Similarly, PB is also a tangent to the circle.
Again, join OC and OD.
Then is an angle in the semicircle and therefore
Since OC is a radius of the given circle, QC has to be a tangent to the circle. Similarly, QD is also a tangent to the circle.
Question 27:
Draw a line segment AB of length . Taking A as centre, draw a circle of radius and taking B as centre, draw another circle of radius . Construct tangents to each circle from the centre of the other circle.
Answer:
To construct: A line segment of length and taking A as centre, to draw a circle of radius and taking B as center, draw another circle of radius . Also, to construct tangents to each circle from the center to the other circle
Steps of Construction:
(a) Bisect BA. Let be the mid-point of BA.
(b) Taking M as center and MA as radius, draw a circle. Let it intersects the given circle at the points P and Q.
(c) Join BP and BQ.Then, BP and BQ are the required two tangents from B to the circle with center A.
(d) Again, let be the mid-point of AB.
(e) Taking M as center and MB as radius, draw a circle. Let it intersects the given circle at the points R and S.
(f) Join AR and AS.Then, AR and AS are the required two tangents from A to the circle with center B
A Line Segment of Length
Justification: Join BP and BQ.
Then being an angle in the semicircle is .
Since AP is a radius of the circle with center A, BP has to be a tangent to a circle with center A. Similarly, BQ is also a tangent to the circle with center A.
Again, join AR and AS.
Then being an angle in the semicircle is .
Since BR is a radius of the circle with center B, AR has to be a tangent to a circle with center B. Similarly, AS is also a tangent to the circle with center B