Two equal parellel chords of the length 24cm .each lie on opposite sides of the centre at a distance of 10cm from each other.Find the diameter of the circle?
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Form two right-angled triangles by drawing a diameter, which bisects both chords. Call the distance from the 24 cm chord to centre, x and the distance from 10 cm chord to centre, y. We are given that x + y = 17 cm.
The right-angled triangle associated with the 24 cm chord has base, x and altitude 12 cm. The hypotenuse of this triangle is the radius of the circle. Call it r for the moment. The other right-angled triangle has base, y, altitude, 5 cm and hypotenuse, r, the radius of a circle.
Using Pythagoras theorem to form two equations:
r^2 = 12^2 + x^2 .. (1) and r^2 = 5^2 + y^2 … (2)
As both expressions equal r^2 we equate the expressions as follows, with some simplifications:
144 + x^2 = 25 + y^2 … (3)
but y = 17 - x, and so substitute this value for y into (3) to get
144 + x^2 = 25 + (17 - x)^2 or with further simplifications:
144 + x^2 = 25 + 289 - 34x + x^2, now subtract x^2 from both sides, leaving a simple equation in x. Do the arithmetic to get x = 5.
The triangle associated with the 24 cm chord is a Pythagorean triplet, a 5, 12, 13 right-angled triangle and you know immediately that r = 13 cm, which is the required answer.
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