Question

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.​

Answer 1

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Plz refer to the attachment for the figure of the question.

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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.

$\huge \mathfrak \blue { \underline {Solution}}$

Let the common chord be AB and P and Q be the centers of the two circles.

∴ AP = 5 cm and AQ = 3 cm.

PQ = 4cm ....given

Now, segment PQ ⊥ chord AB

∴ AR = RB =

$\frac{1}{2} \: ab$ ....perpendicular from center to the chord, bisects the chord

Let PR = xcm, so RQ = ( 4 − x )cm

In △ARP,

AP ² = AR ² + PR ²

AR ² = 5 ² − x ². ...(1)

In △ARQ,

AQ ² = AR ² + QR ²

AR ² = 3 ² − ( 4 − x ) ² ...(2)

∴ 5 ² − x ² =3 ² − ( 4 − x ) ² ....from (1) & (2)

25 − x ² = 9 − ( 16 − 8x + x ² )

25 − x ² = −7 + 8x − x ²

32=8x

∴ x=4

Substitute in eq (1) we get,

AR ² = 25 − 16 = 9

∴ AR = 3 cm.

∴ AB = 2 × AR = 2 × 3

∴ AB = 6cm.

So, length of common chord AB is 6cm .

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I hope my answer helps you....

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Answer 2

Answer:

Here is the answer

Step-by-step explanation:

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