In the adjoining diagram, AB = x cm, BC = y cm and x - y = 7 cm. Area of △ ABC = 30 cm². The length of AC is :

1. 10 cm

2. 12 cm

3. 13 cm

4. 15 cm

By formula,

Area of triangle = $\dfrac{1}{2} \times \text{ base} \times \text{height}$

⇒ 30 = $\dfrac{1}{2} \times BC \times AB$

⇒ 30 = $\dfrac{1}{2} \times y \times x$

⇒ 30 = $\dfrac{xy}{2}$

⇒ xy = 60 ……..(1)

Given,

⇒ x - y = 7

⇒ x = 7 + y ……..(2)

Substituting value of x from equation (2) in (1), we get :

⇒ (7 + y)y = 60

⇒ 7y + y² = 60

⇒ y² + 7y - 60 = 0

⇒ y² + 12y - 5y - 60 = 0

⇒ y(y + 12) - 5(y + 12) = 0

⇒ (y - 5)(y + 12) = 0

⇒ y - 5 = 0 or y + 12 = 0

⇒ y = 5 or y = -12.

Since, side cannot be negative,

∴ y = 5 cm.

Substituting value of y in equation (2), we get :

⇒ x = 7 + y = 7 + 5 = 12 cm.

In right angle triangle ABC,

By pythagoras theorem,

⇒ AC² = AB² + BC²

⇒ AC² = x² + y²

⇒ AC² = 12² + 5²

⇒ AC² = 144 + 25

⇒ AC² = 169

⇒ AC = $\sqrt{169}$ = 13 cm.

Hence, Option 3 is the correct option.