In a triangle ABC, P is the midpoint of BC. If AB = (2x + 4) cm, AC = 6 cm and AP⊥ BC, then the value of x is:

Given:

In a triangle ABC, P is the midpoint of BC.

AB = (2x + 4) cm

AC = 6 cm

AP ⊥ BC

Formula used:

In a right-angled triangle, Pythagoras theorem: a² + b² = c²

Calculation:

Given AB = (2x + 4) cm, AC = 6 cm, and AP ⊥ BC, AP is the altitude.

Since P is the midpoint of BC and AP is perpendicular to BC, triangle APB and triangle APC are right-angled at P.

Using Pythagoras theorem in triangle APC:

AP² + PC² = AC²

Since P is the midpoint of BC, PC = BC/2.

Let BC = 2y, then PC = y.

Given AC = 6 cm, we have:

AP² + y² = 6²

AP² + y² = 36

Using Pythagoras theorem in triangle APB:

AP² + PB² = AB²

Since P is the midpoint of BC, PB = y.

Given AB = (2x + 4) cm, we have:

AP² + y² = (2x + 4)²

AP² + y² = 4x² + 16x + 16

Equating the two equations for AP² + y²:

36 = 4x² + 16x + 16

Solving for x:

36 - 16 = 4x² + 16x

20 = 4x² + 16x

4x² + 16x - 20 = 0

Dividing by 4:

x² + 4x - 5 = 0

Factoring the quadratic equation:

(x + 5)(x - 1) = 0

⇒ x = -5 or x = 1

Since x is a length, it must be positive.

⇒ x = 1

∴ The correct answer is option (3).