The other sides are radii, therefore OA= OC and OB= OD. Let us consider two equal chords in the circle with centre O. Join the end points of the chords with the centre to get the triangles Δ AOB and Δ OCD, chord AB = chord CD (because the given chords are equal). Now we are going to discuss another property. Instead of a single chord we consider two equal chords. Draw three circles passing through the points P and Q, where PQ = 4cm.Ģ. Find the distance of the chord from the centre.Ģ. The radius of the circle is 25 cm and the length of one of its chord is 40cm. (1) (Perpendicular drawn from centre to chord bisect it)ġ. Given : Chord AB of the outer circle cuts the inner circle at C and D. In the concentric circles, chord AB of the outer circle cuts the inner circle at C and D as shown in the diagram. “In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides”.Īpplication of this theorem is most useful in this unit. One of the most important and well known results in geometry is Pythagoras Theorem. Let AB be the chord and C be the mid point of AB Theorem 7 The perpendicular from the centre of a circle to a chord bisects the chord.Ĭonverse of Theorem 7 The line joining the centre of the circle and the midpoint of a chord is perpendicular to the chord.įind the length of a chord which is at a distance of 2√11 cm from the centre of a circle of radius 12cm. This argument leads to the result as follows. RHS criterion tells us that Δ AOC and Δ BOC are congruent. ∠ OCA = ∠ OCB = 90 ° ( OC ⊥ AB ) and OA = OB is the radius of the circle. Here, easily we get two triangles Δ AOC and Δ BOC (Fig.4.56).Ĭan we prove these triangles are congruent? Now we try to prove this using the congruence of triangle rule which we have already learnt. Perpendicular from the Centre to a ChordĬonsider a chord AB of the circle with centre O. Now, we are going to discuss some properties based on chords of the circle.Ĭonsidering a chord and a perpendicular line from the centre to a chord, we are going to see an interesting property.ġ. Using all the properties of these, we get some standard results one by one. Recently we have seen a new member circle. Circular segment - the part of the sector that remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary.In this chapter, already we come across lines, angles, triangles and quadrilaterals.The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be ( cos θ, sin θ), and then using the Pythagorean theorem to calculate the chord length: \displaystyle See also The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure). The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle. The chord function is defined geometrically as shown in the picture. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits after the integer part. In the second century AD, Ptolemy of Alexandria compiled a more extensive table of chords in his book on astronomy, giving the value of the chord for angles ranging from 1 / 2 to 180 degrees by increments of 1 / 2 degree. The first known trigonometric table, compiled by Hipparchus, tabulated the value of the chord function for every 7 1 / 2 degrees. Chords were used extensively in the early development of trigonometry.
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