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Grade 9 → Math → Circles → 9.2-Perpendicular from the Centre to a Chord
- Understand perpendicular from the centre to a chord.
Theorem 9.3 : The perpendicular from the centre of a circle to a chord bisects the chord.:
Theorem 9.4 : The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.:
Theorem 9.5 : Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres):
Theorem 9.6 : Chords equidistant from the centre of a circle are equal in length.:
Solved Example: 9-2-01
Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.
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Solution:
6 cm. First show that the line joining centres is perpendicular to the radius of the smaller circle and then that common chord is the diameter of the smaller circle.
Solved Example: 9-2-02
If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.
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Solution:
If AB, CD are equal chords of a circle with centre O intersecting at E, draw perpendiculars OM on AB and ON on CD and join OE. Show that right triangles OME and ONE are congruent.
Solved Example: 9-2-03
If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.
Solved Example: 9-2-04
If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD (see Fig. 9.12).
Solved Example: 9-2-05
Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6m each, what is the distance between Reshma and Mandip?
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Solution:
Represent Reshma, Salma and Mandip by R, S and M respectively.
Let KR = x m (see figure).
Area of ∆ ORS = $\dfrac{1}{2}$ x $\times$ 5 .
Also , area of ∆ ORS = $\dfrac{1}{2}$ RS×OL= 2 × 6 ×4.
Find x and hence RM.
Solved Example: 9-2-06
A circular park of radius 20m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone.
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Solution:
Use the properties of an equilateral triangle and also Pythagoras Theorem.
