Question

If a diagonal of a rectangle is thrice its smaller side, then its length and breadth are in the ratio

(a) 3 : 1
(b) $\sqrt{3}:1$
(c) $\sqrt{2}:1$
(d) $2\sqrt{2}:1$

Answer 1

(d) $2\sqrt{2}$ : 1
Let us assume that the length of the smaller side of the rectangle, i.e., BC be x and length of the larger side , i.e., AB be y.
It is given that the length of the diagonal is three times that of the smaller side.
Therefore, diagonal = 3x = AC
Now, applying Pythagoras theorem, we get:
(Diagonal)² = (Smaller side)² + (Larger side)²

(AC)² = (AB)² + (BC)²
(3x)² = (x)² + (y)²
9x² = x² + y²
8x² = y²
Now, taking square roots of both sides, we get:
$2\sqrt{2}$ x = y
or, $\frac{y}{x}=\frac{2\sqrt{2}}{1}$

Thus, the ratio of the larger side to the smaller side = $2\sqrt{2}$ : 1