Maths lovers blog

Before going to the topic remember these terms, Chord:- A line segment connecting two points on a curve.  Diameter:- A straight line going through the center of a circle connecting two points on the circumference. Perpendicular Line:- Lines that are at right angles ( 90° ) to each other. When a perpendicular line is drawn from the center of the circle to the chord. Then the perpendicular line bisects (divide into 2 halves) the chord. Lets take both of them (AOP & BOP) as right angle triangle and we can see that from point O the radius is same. Here the distance of OP is same in both of the triangles. On the other hand, the angle AOP and BOP are also same. Hence, by using the property of S.A.S the length AP and BP are same and thus, are divided equally (bisect) by the chord. For its derivation lets proceed forward, Given a circle with centre O and AC = BC To  Prove: OC ⊥ AB Construction: Join points O, A and O, B. Proof: In Δ’s OCA and OCB OA =...

Non-collinear points :- Non-collinear means simply ‘does not lies on a common line’, so it is a property of a set of points (at least two) of a geometry. We join these three points with line. After joining these points when we draw a perpendicular bisector to each of these lines. When perpendicular bisectors are drawn they cut each other and from that point the length of each line is equal. This line is the radius of the circle,hence a circle drawn on these non-collinear points. When there are three non-collinear points (not lying on a single line), then only one circle can be drawn on which all these points lie. But, if we try to draw any other circle on these points then it would also be of the  same  radius. Its derivation is given below, Given : Three non collinear points P, Q and R To prove : There is one and only one circle passing through the points P, Q and R. Construction:  Join PQ and QR. Draw perpendicular bisectors AB of PQ ...