Today is our topic of discussion Some Constructions Involving Circles .
Some Constructions Involving Circles
Construction 5.
Draw such a circle which passes through two definite points and whose centre lies on a definite straight line.
A and B are the two fixed points, PQ is a fixed straight line. Construct such a circle which passes through the points A and B and whose centre lies on the straight line PQ.
Drawing:
Step 1. Join A, B.
Step 2. Construct the perpendicular bisector CD of line segment AB.
Step 3. The line segment CD intersects the line PQ at the point O.
Step 4. Taking O as centre draw the circle of radius OA or OB, ABNM is the desired circle.
Proof:
The line segment C’Dis perpendicular bisector of AB. Therefore, any point on CD is of equal distance from A and B. By construction, the point O lies on CD and PQ.
Again, since OA and OB are equal so the circle drawn at the centre O with the radius OA or OB will pass through the points A and B and the centre O will lie on the line segment PQ… the circle drawn with O as centre and OA or OB is the required circle.
Construction 6.
With the radius equal to a definite line segment, construct a circle which passes through two definite points. A and B are the definite points and r is the length of the definite line segments. Construct such a circle which passes through A and B and whose radius is equal to r.
Drawing:
Step 1. Join A and B.
Step 2. Draw two segments of the two circles of radius by centering A and B and taking r as radius on both sides of the line AB. The two pairs of segments of the two circles intersect at P and Q on the two sides of the line AB respectively.
Step 3. Taking P as centre and PA as radius, draw the circle ABC.
Step 4. Again taking Q as centre and QA as radius, draw the circle ABD.
Step 5. Then each of ABC and ABD is the required circle.
Proof:
PA PB = r… The drawn circle ABC with centre P and radius PA = or PB that passes through the points A and B and its radius is PA = r.
Again QA QB = r… The drawn circle ABD with centre Q and radius QA or QB passes through the points A and B and its radius is QA = r. Each of the two circles ABC and ABD is the required circle.
Construction 7. Construct a circle which touches a definite point of a definite circle and passes through a definite point outside the circle
Let a circle be given with centre C, P be a definite point on that circle and Q be a definite point outside that circle. Draw a circle which touches the circle at P and passes through the point Q.
Drawing:
Step 1. Join P, Q.
Step 2. Draw the perpendicular bisector AB of PQ.
Step 3. Join C, P.
Step 4. Extended line segment CP intersects AB at the point O.
Step 5. Taking O as centre, draw the circle with radius equal to OP. The resulting circle PQR is the required circle.
Proof:
Join O, Q. The line segment AB or the line segment OB is the perpendicular bisector of PQ… OP=OQ. So the circle of radius OP and centre O will pass through Q.
Again the point P lies on the given circle and on the constructed circle and also the line joining the centres of the two circles. i.e. the two circles intersect at the point P. So the two circles touch each other at P.
Therefore, the circle drawn with O as the centre and OP as the radius is the required circle.
Construction 8.
Construct a circle which touches a definite point on a definite straight line and passes through a point. Suppose A is a definite point on the straight line AB and P be a point not lying on the line AB. To draw a circle which touches the line AB at A passes through the point P.
Drawing:
Step 1. Draw the perpendicular AC at the point A on the line AB.
Step 2. Join P, A and construct its perpendicular bisector QR.
Step 3. The lines QR and AC intersect at O.
Step 4. Taking O as centre draw the circle APS with radius OA. Then APS is the required circle.
Proof:
Join O, P. The point O lies on QR that is perpendicular bisector of AP.
OA = OP. The circle with centre O and radius OA passes through the points P.
Again, OA passing through A is a perpendicular to the line AB.
So the circle touches line AB at the point A.
Taking O as the centre and OA as the radius, the drawn circle is the required circle.
Analysis: Since the circle is required to touch a definite line at a definite point, so, that line has to be tangent to the circle at that definite point and this tangent has to be the diameter of the circle. Since the definite point on the line and the definite external point both are required to lie on the circle .
See more:
