XAB and XCD are two secants to a circle. If XA = 18 cm, AB = 22 cm and XC = 24 cm, then find the value of XD (in cm).

Question

Accepted Answer

Given two secants XAB and XCD intersecting a circle from point X outside the circle, the secant segment lengths satisfy the relation: XA * AB = XC * XD.

Step 1: Calculate XA * AB
XA = 18 cm, AB = 22 cm
XA * AB = 18 * 22 = 396

Step 2: Use the relation to find XD
XC * XD = 396
Given XC = 24 cm
So, 24 * XD = 396
XD = 396 / 24 = 16.5 cm

However, this contradicts the options given, so let's re-check the problem interpretation.

The correct formula for two secants from an external point X is:
(XA + AB) * XA = (XC + CD) * XC

Given:
XA = 18 cm
AB = 22 cm
XC = 24 cm
Let XD = d cm

So,
(XA + AB) * XA = (XC + XD) * XC
(18 + 22) * 18 = (24 + d) * 24
40 * 18 = (24 + d) * 24
720 = 24 * (24 + d)
720 = 576 + 24d
24d = 720 - 576 = 144
d = 144 / 24 = 6 cm

This also does not match options.

Re-examining the problem, the standard secant theorem states:
(XA) * (XB) = (XC) * (XD)
Where XB = XA + AB
So,
XA = 18 cm
AB = 22 cm
XB = 18 + 22 = 40 cm
XC = 24 cm
XD = ?

Apply the theorem:
XA * XB = XC * XD
18 * 40 = 24 * XD
720 = 24 * XD
XD = 720 / 24 = 30 cm

Therefore, the correct value of XD is 30 cm.

Check options: 30 is option 3.

Final answer: 30 cm.

Answer

Answer

Answer