Math, asked by indusati4317, 6 months ago

Two circle of radii 5 cm and 3 cm intersect two points and the distance between their centers is 4cn find the length vof the common CHORD

Answers

Answered by ExᴏᴛɪᴄExᴘʟᴏʀᴇƦ
9

Answer

  • The length of the common chord is 6 cm

Explanation

Given

  • Radius of one circle is 5 cm
  • Radius of the other one is 3 cm
  • The distance between their centers will be 4 cm

To Find

  • Length of the common chord

Solution

  • OA = 5 cm
  • AO' = 3 cm
  • OO' = 4 cm
  • AB = 2AC

In ∆AOC

→ AO² = AC² + OC²

→ 5² = (4-x)² + OC²

→ 25 = 16+x² - 8x + OC²

OC = 9-x²+8x -eq(1)

In ACO'

→ AO'² = AC²+O'C²

3² = AC²+x² -eq(2)

Equating eq(1) & eq(2)

→ 9-x²+8x = 9-x²

Cancelling the like terms

→ 8x = 0

→ x = 0

Substituting this in eq(1)

→ AC = 3 cm

Length of the chord will be

→ AB = 2AC

→ AB = 3×2

→ AB = 6 cm

Attachments:
Answered by ADITYABABBAR
1

Let two circles with centres O and O’ intersect each other at points A and B. On joining A and B, AB is a common chord.

Let two circles with centres O and O’ intersect each other at points A and B. On joining A and B, AB is a common chord.Radius OA = 5 cm, Radius O’A = 3 cm,

Let two circles with centres O and O’ intersect each other at points A and B. On joining A and B, AB is a common chord.Radius OA = 5 cm, Radius O’A = 3 cm,Distance between their centers OO’ = 4 cm

Let two circles with centres O and O’ intersect each other at points A and B. On joining A and B, AB is a common chord.Radius OA = 5 cm, Radius O’A = 3 cm,Distance between their centers OO’ = 4 cmIn triangle AOO’,

52 = 42 + 32

52 = 42 + 32 25 = 16 + 9

52 = 42 + 32 25 = 16 + 9 25 = 25

52 = 42 + 32 25 = 16 + 9 25 = 25Hence AOO’ is a right triangle, right angled at O’.

52 = 42 + 32 25 = 16 + 9 25 = 25 Hence AOO’ is a right triangle, right angled at O’.Since, perpendicular drawn from the center of the circle bisects the chord.

52 = 42 + 32 25 = 16 + 9 25 = 25 Hence AOO’ is a right triangle, right angled at O’.Since, perpendicular drawn from the center of the circle bisects the chord.Hence O’ is the mid-point of the chord AB. Also O’ is the centre of the circle II.

Therefore length of chord AB = Diameter of circle II

Length of chord AB = 2 x 3 = 6 cm.

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