Question

Two circle is radii 3cms and 5cms intersect each other at a point. If the distance between their point are 4cms. Find the length of the chord connecting​

Answer 1

Please refer the attached diagram to get proper reference with the answer.

⇒ Correct Question:

Two circle is radii 3cm and 5cm intersect each other at a point. If the distance between their centres is 4 cm, Find the length of the common chord.

⇒ Given:

Two circles have the radii 3 cm and 5 cm. [AX = 5 cm and CX = 3 cm]

These circles intersect at a point. [Point B]

The distance between the centre of these circles is 4 cm. [AC = 4cm]

⇒ To Find:

The length of the common chord.

⇒ Solution:

At first, we can write down the names of the elements we would be using in this question:

Chord = XY

Line segment connecting centres = AC

5 cm radius = AX

3 cm radius = CX

As per the chord bisecting theorem,

The line segment joining the centers of the two circles is perpendicular to the chord and bisects it into two equal halves.

So, AC⊥XY and and XB = BY.

We know that AC is 4 cm.

Let AB be x, so BC is 4 - x.

In △AXB,

AX² = XB² + AB²

5² = XB² + x²

XB² = 5² - x² ---------- (1)

In △BXC,

CX² = BX² + BC²

3² = BX² + (4 - x)²

BX² = 3² - (4 - x)² ---------- (2)

Combining Equations (1) and (2):

• 5² - x² = 3² - (4 - x)²

Expanding the statement:

• 25 - x² = 9 - 16 - 8x + x²

Solving:

• 25 - x² = -7 - 8x + x²

Moving -7 to the LHS and cancelling x² terms:

• 32 = 8x

Therefore, x:

• $\sf{x=\dfrac{32}{8}}$

• x = 4

Now, giving the value of x in the equations (1) and (2):

5² - x² = XB² ---------- (1)

25 - 16 = XB²

XB² = 9

XB = √9 = 3 cm

As XY = 2 x XB:

XY = 2 x 3

XY = 6 cm

Hence the length of the chord XY is 6 cm.