(i)

(ii)

(iii)

(iv)

(v)

Find the unknown length x in each of the following figures.

We know that,

If two chords of a circle intersect internally, then the products of the length of segments are equal.

(i) PA × PB = PC × PD

5 × 5.6 = 3.5 × x

x = $\dfrac{28}{3.5}$

x = 8 cm.

Hence, x = 8 cm.

(ii) PA × PB = PC × PD

x × 9 = 8.1 × 5

x = $\dfrac{40.5}{9}$

x = 4.5 cm

Hence, x = 4.5 cm.

We know that,

If two chords of a circle intersect externally, then the products of the length of segments are equal.

(iii) PA × PB = PC × PD

⇒ PB = PA + AB = 7 + 9 = 16

⇒ PD = PC + CD = 8 + x

⇒ 7 × 16 = 8 × (8 + x)

⇒ 112 = 64 + 8x

⇒ 8x = 112 - 64

⇒ x = $\dfrac{48}{8}$

⇒ x = 6 cm.

Hence, x = 6 cm.

(iv) We know that,

If a chord and a tangent intersect externally, then the product of lengths of the segments of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection.

∴ PT² = AP × BP

x² = 4.5 × 18

x² = 81

x = 9 cm.

Hence, x = 9 cm.

(v) We know that,

If a chord and a tangent intersect externally, then the product of lengths of the segments of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection.

∴ PT² = AP × BP

⇒ BP = AP + AB = x + 10

⇒ 12² = x × (x + 10)

⇒ 144 = x² + 10x

⇒ x² + 10x - 144 = 0

⇒ x² + 18x - 8x - 144 = 0

⇒ x(x + 18) - 8(x + 18) = 0

⇒ (x - 8)(x + 18) = 0

⇒ (x - 8) = 0 or (x + 18) = 0      [Using Zero-product rule]

⇒ x = 8 or x = -18

∴ x = 8 because length cannot be negative.

Hence, x = 8 cm.