The length of a rectangle is 1 metre more than its breadth. Its area is 3 3 4 square metres. What are its length and breadth?

Let the length and breadth of the rectangle be x m and y m . Given that the length of a rectangle is 1 metre more than its breadth. So, x = y + 1 ⇒ x - y = 1 -----(i) Also, it is given that the area of the rectangle is 3 3 4 square metres. So, x y = 3 3 4 ⇒ x y = 15 4 -----(ii) Now, we know that ( x + y ) 2 - ( x - y ) 2 = 4 x y ⇒ ( x + y ) 2 - ( 1 ) 2 = 4 × 15 4 ⇒ ( x + y ) 2 - 1 = 15 ⇒ ( x + y ) 2 = 15 + 1 = 16 ⇒ x + y = 16 = 4 m .-----(iii) Adding (i) and (ii), we get, x - y + x + y = 1 + 4 ⇒ 2 x = 5 ⇒ x = 5 2 = 2 1 2 m . Substituting the value of x in (iii), we get, x + y = 4 ⇒ y = 4 - x = 4 - 5 2 ⇒ y = 8 - 5 2 = 3 2 = 1 1 2 m . Therefore, the length and breadth of the rectangle are 2 1 2 m and 1 1 2 m , respectively.

The hypotenuse of a right triangle is 6 1 2 centimetres and its area is 7 1 2 square centimetres. Calculate the lengths of its perpendicular sides.

Let the lengths of the perpendicular sides be x cm (bigger side) and y cm (smaller side). Given that the hypotenuse of a right triangle is 6 1 2 cm . So, using Pythagoras theorem on △ A B C , we get, x 2 + y 2 = 6 1 2 2 = 13 2 2 = 169 4 ⇒ x 2 + y 2 = 169 4 -----(i) Also, given that the area of the triangle is 7 1 2 cm 2 . So, the area of the triangle is = 1 2 × base × height = x y 2 cm 2 . Hence, we have x y 2 = 7 1 2 = 15 2 ⇒ x y = 15 -----(ii) Now, from (i), we get, x 2 + y 2 = 169 4 ⇒ x + y 2 - 2 x y = 169 4 ⇒ ( x + y ) 2 - 2 × 15 = 169 4 ⇒ ( x + y ) 2 - 30 = 169 2 ⇒ ( x + y ) 2 = 169 4 + 30 = 169 + 120 4 = 289 4 ⇒ x + y = 289 4 = 17 2 ⇒ x + y = 17 2 -----(iii) Again, from (i), we get, x 2 + y 2 = 169 4 ⇒ x - y 2 + 2 x y = 169 4 ⇒ ( x - y ) 2 + 2 × 15 = 169 4 ⇒ ( x - y ) 2 + 30 = 169 4 ⇒ ( x - y ) 2 = 169 4 - 30 = 169 - 120 4 = 49 4 ⇒ x - y = 49 4 = 7 2 ⇒ x - y = 7 2 -----(iv) Adding (iii) and (iv), we get, x + y + x - y = 17 2 + 7 2 ⇒ 2 x = 17 + 7 2 = 24 2 = 12 ⇒ x = 12 2 = 6 cm . From (iii), we get, x + y = 17 2 ⇒ y = 17 2 - x = 17 2 - 6 = 17 - 12 2 = 5 2 = 2 1 2 cm . Therefore, the lengths of the perpendicular sides of the triangle are 6 cm and 2 1 2 cm .