Circles are the first approximation to the orbits of planets and of their moons, to the movement of electrons in an atom, to the motion of a vehicle around a curve in the road, and to the shapes of cyclones and galaxies. The theoretical importance of circles is reflected in the amazing number and variety of situations in science where circles are used to model physical phenomena. Students traditionally learn a greater respect and appreciation of the methods of mathematics from their study of this imaginative geometric material. The logic becomes more involved − division into cases is often required, and results from different parts of previous geometry modules are often brought together within the one proof. They clearly need to be proven carefully, and the cleverness of the methods of proof developed in earlier modules is clearly displayed in this module. The theorems of circle geometry are not intuitively obvious to the student, in fact most people are quite surprised by the results when they first see them. Tangents are introduced in this module, and later tangents become the basis of differentiation in calculus. Lines and circles are the most elementary figures of geometry − a line is the locus of a point moving in a constant direction, and a circle is the locus of a point moving at a constant distance from some fixed point − and all our constructions are done by drawing lines with a straight edge and circles with compasses. Chords which are having the same length are at equidistant from the center of the circle and the chords which are at equidistant from the center are of equal length.Most geometry so far has involved triangles and quadrilaterals, which are formed by intervals on lines, and we turn now to the geometry of circles.The line that bisects the chord from the center of the circle is perpendicular to the chord and that line is referred to as the perpendicular bisector of the chord.A chord of a circle is a line that joins the two points on the boundary of the circle and it is called a secant if the chord is extended external to the circle.A circle is a combination of points in a place that is at equidistant from the center O. OX=OY(Distance from the center are equal)īy SSS rule, ∆AOX≅∆COY and by RHS criterion, ∠ AOX = ∠ COY=90°, Regarding the details in the above theorem, Proof: AB and CD are two chords of a circle with center O, Both chords are at an equal distance from the center O, we have to prove that AB=CD Theorem 6: Two Chords of a circle which are at the same distance from the Center O are equal in length (Converse of Theorem 5) This proves that OX=OY, so two chords are at equal distance from the center O X bisects the AB as it is perpendicular to AB and as the same Y bisects the CD as it is perpendicular to CDīy SSS rule ∆AOX≅∆COY and by RHS criterion, ∠ AOX = ∠ COY=90°, The distance of AB from the center is marked as X and The distance of CD from the center is marked as Y Proof: AB and CD are two chords of a circle with center O and we have to prove that AB and CD are at equal distances from center O Theorem 5: Both the Equal chord of a circle are at equal distance from the center of a circle
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