Question

A pair of adjacent sides of a rectangle is in the ratio 5:12. If the length of the diagonal is 26 cm . Find the length of side and perimeter of rectangle​

Answer 1

Solution :-

Ratio of sides of a rectangle = 5 : 12

Let the constant ratio be x

So, sides of a rectangle be 5x and 12x

i.e Dimensions of rectangle :

• Length = 12x
• Breadth = 5x

Length of the diagonal = 26 cm

Consider the triangle formed by diagonal and adjacent sides.

In a rectangle all angles are right angles

Therefore, the triangle formed by diagonal and adjacent sides is a Right angled triangle.

By pythagoras theorem

(Length of the rectangle)² + (Breadth of the reactangle)² = (Diagonal of the rectangle)²

⇒ ( 12x )² + ( 5x )² = 26²

⇒ 144x² + 25x² = 676

⇒ 169x² = 676

⇒ x² = 676/169

⇒ x² = 4

⇒ x = ± √4

⇒ x = ± 2

Neglecting x = - 2

⇒ x = 2

Length of the rectangle = 12x = 12 * 2 = 24 cm

Breadth of the rectangle = 5x = 5 * 2 = 10 cm

Perimeter of the rectangle = 2(l + b)

= 2(24 + 10)

= 2 * 34

= 68 cm

Hence, lengths of the sides are 24 cm, 10 cm and the perimeter of rectangle is 68 cm.

Answer 2

Answer:

Given:

• Ratio of sides of a rectangle = 5 : 12
• Length of Diagonal = 26 cm.

$\rule{350}{1}$

Let unit be x.

• Length = 12x
• Breadth = 5x

Now, we know that all the angles of a rectangle are right angles. So, the triangle formed by diagonal and its adjacent sides is also Right angled triangle.

Now, By Pythagoras theorem

⇒ (Diagonal)² = (Length)² + (Breadth)²

⇒ (26)² = (12x)² + (5x)²

⇒ 676 = 144x² + 25x²

⇒ 676 = 169x²

⇒ x² = 676/169

⇒ x² = 4

⇒ x = √4

⇒ x = ±2

We neglect -2 because we know dimension cannot be negative.

So, x = 2

• Length = 12x = 12 × 2 = 24 cm
• Breadth = 5x = 5 × 2 = 10 cm

$\rule{350}{1}$

Now, Perimeter = 2(l + b)

⇒ Perimeter = 2(24 + 10)

⇒ Perimeter = 2(34)

⇒ Perimeter = 68 cm

$\rule{350}{2}$

Hence,

• Perimeter of Rectangle = 68 cm
• Length of Rectangle = 24 cm
• Breadth of Rectangle = 10 cm