You are given just a compass and a ruler which does not show any measurement on it. You see a circle on the cartesian coordinate plane. Call this circle ‘C’. You are required to construct a circle (call this circle ‘D’) concentric to circle C and with radius which is √2 times the radius of C. You do not have the value of ‘r’, the radius of C. Study the following steps carefully. Remember that the only thing you can do with the ruler provided is just draw a straight line. What is the first step to construct the circle ‘D’?
You are given just a compass and a ruler which does not show any measurement on it. You see a circle on the cartesian coordinate plane. Call this circle ‘C’. You are required to construct a circle (call this circle ‘D’) concentric to circle C and with radius which is √2 times the radius of C. You do not have the value of ‘r’, the radius of C. Study the following steps carefully. Remember that the only thing you can do with the ruler provided is just draw a straight line. What is the first step to construct the circle ‘D’?
The correct option is A To determine the centre of the circle.
Since we need to construct a concentric circle, we need the centre of these concentric circles first. So finding out the centre would be first obvious thing to start with. The idea is, once we set up the centre of the given circle ‘C’, we construct two radii which are perpendicular to each other, as shown.
Since OP and OQ are perpendicular, by Pythagoras theorem, PQ = r√2
Now we have the length of the radius and the centre, we are good to go.
Now let us see how to proceed with the construction.
Step I : Draw a random chord of the circle.
Step II : Bisect this chord perpendicularly. Since the radius is the perpendicular bisector of all chords of a circle, this perpendicular bisector when extended so that it cuts the circle twice, becomes the diameter. Let this line cut the circle at P and Y.
Step III : Now perpendicularly bisect this diameter. Even this bisector also forms the diameter as this line is equal to diameter PY. The point of intersection of these diameters gives us the centre, O.
Step IV : Join PQ. This radius is r√2. So with radius PQ and centre O, draw the required circle.
To determine the centre of the circle.
Since we need to construct a concentric circle, we need the centre of these concentric circles first. So finding out the centre would be first obvious thing to start with. The idea is, once we set up the centre of the given circle ‘C’, we construct two radii which are perpendicular to each other, as shown.
Since OP and OQ are perpendicular, by Pythagoras theorem, PQ = r√2
Now we have the length of the radius and the centre, we are good to go.
Now let us see how to proceed with the construction.
Step I : Draw a random chord of the circle.
Step II : Bisect this chord perpendicularly. Since the radius is the perpendicular bisector of all chords of a circle, this perpendicular bisector when extended so that it cuts the circle twice, becomes the diameter. Let this line cut the circle at P and Y.
Step III : Now perpendicularly bisect this diameter. Even this bisector also forms the diameter as this line is equal to diameter PY. The point of intersection of these diameters gives us the centre, O.
Step IV : Join PQ. This radius is r√2. So with radius PQ and centre O, draw the required circle.
