The lengths of two sides of a right triangle containing the right angle differ by 2 cm. If the area of the triangle is 24 cm², find the perimeter of the triangle.
Mensuration
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Question

The lengths of two sides of a right triangle containing the right angle differ by 2 cm. If the area of the triangle is 24 cm2, find the perimeter of the triangle.

KnowledgeBoat · Accepted Answer

ABC is a right angled triangle with a right angle at B. The area of the triangle is 24 cm2. Let the lengths of BC and AB be x and y, respectively. Given, The difference between the two perpendicular sides is 2 cm. x - y = 2 ∴ y = x - 2 Area = × base × height ⇒ 24 = × BC × AB ⇒ 24 = × x × (x - 2) ⇒ x × (x - 2) = 48 ⇒ x2 - 2x = 48 ⇒ x2 - 2x - 48 = 0 ⇒ x2 - 8x + 6x - 48 = 0 ⇒ x(x - 8) + 6(x - 8) = 0 ⇒ (x - 8)(x + 6) = 0 ⇒ (x - 8) = 0 or (x + 6) = 0 ⇒ x = 8 or x = -6 Since length cannot be negative, ∴ x = 8 cm. y = x - 2 = 8 - 2 = 6 cm. Thus, AB = 6 cm and BC = 8 cm. By using Pythagoras theorem, Hypotenuse2 = Base2 + Height2 ⇒ AC2 = BC2 + AB2 ⇒ AC2 = 82 + 62 ⇒ AC2 = 64 + 36 ⇒ AC2 = 100 ⇒ AC = ⇒ AC = 10 cm. Perimeter of a triangle = Sum of all the sides of a triangle = AB + BC + AC = 10 + 8 + 6 = 24 cm. Hence, perimeter of a triangle = 24 cm.

Mathematics

The lengths of two sides of a right triangle containing the right angle differ by 2 cm. If the area of the triangle is 24 cm², find the perimeter of the triangle.

Mensuration

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