Question

two circles of radii 5cm and 3 cm intersect at two points and the distance between their chords is 4cm.Find the length of the common chord

Answer 1

Let the radius of the two circles be 5 cm and 3 cm respectively whose centre’s are O and O'
Hence OA = OB = 5 cm
O'A = O'B = 3 cm
OO' is the perpendicular bisector of chord AB.
Therefore, AC = BC
Given OO' = 4 cm
Let OC = x
Hence O'C = 4 − x
In right angled ΔOAC, by Pythagoras theorem
OA ^2= OC^2 + AC^2
⇒ 5^2= x^2 + AC^2
⇒ AC^2 = 25 − x^2 à (1)
In right angled ΔO'AC, by Pythagoras theorem
O'A ^2= AC^2 + O'C^2
⇒ 3^2 = AC^2+ (4 – x)^2
⇒ 9 = AC^2 + 16 + x^2− 8 x
⇒ AC^2 = 8x − x^2 − 7 à (2)
From (1) and (2), we get
25 − x^2 = 8x − x^2 − 7
8x = 32
Therefore, x = 4
Hence the common chord will pass through the centre of the smaller circle, O' and hence, it will be the diameter of the smaller circle.
AC^2= 25 − x^2
= 25 − 4^2
= 25 − 16 = 9
Therefore, AC = 3 m
Length of the common chord, AB = 2AC = 6 m

Answer 2

Hi there!

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For solutions, Refer to the attached picture.
Regrets for handwriting _/\_

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Let's see some related topics :

⚫ Circle : The collection of all the points, which are at a fixed distance from a fixed point in a plane, is called a circle.

⚫ Radius : A line joining the centre to the Circumference of the circle, is called radius of a circle.

⚫ Secant : A line intersecting a circle at any two points, is called secant.

⚫ Diameter : A chord passing through the point of the circle, is called diameter. It is the longest chord.

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