In the given circle, chords AB and CD intersect internally at P. If $$\overline{\text{CP}}$$​ = 3 cm and $$\overline{\text{DP}}$$​ = 8 cm, and the numerical values of the lengths of $$\overline{\text{AP}}$$​ and $$\overline{\text{BP}}$$​ are both natural numbers, then which of the following options CANNOT be the length of $$\overline{\text{AB}}$$​?

Correct option is B

Given:

Chords AB and CD intersect internally at point P.

CP = 3 cm

DP = 8 cm

The lengths of AP and BP are natural numbers.

Concept Used:

The intersecting chords theorem states:

(AP) × (BP) = (CP) × (DP)

Solution:

We are tasked with finding which of the following options CANNOT be the length of AB:

A) 14 cm

B) 20 cm

C) 25 cm

D) 10 cm

(AP) × (BP) = 3 × 8 = 24

Let AP = x and BP = y, where both x and y are natural numbers. From the equation:

x × y = 24

The possible pairs of natural numbers x and y that satisfy this equation are:

(1, 24), (2, 12), (3, 8), (4, 6)

Thus, the possible lengths of AB (which is AB = AP + BP = x + y) are:

1 + 24 = 25, 2 + 12 = 14, 3 + 8 = 11, 4 + 6 = 10

The possible lengths for AB are 25, 14, 11, and 10. From the options given, 20 cm does not match any of these values.

Thus, the correct answer is (B) 20 cm