Question

ABCD is a rhombus with diagonals BD and AC equal to 12 cm and 16 cm respectively. The value of cosec x is ______.

पर्याय

• `1 2/3`

• `3 1/3`

• `1/2`

• `1 1/3`

Answer 1

ABCD is a rhombus with diagonals BD and AC equal to 12 cm and 16 cm respectively. The value of cosec x is `bbunderline(1 2/3)`.

Explanation:

The diagonals of a rhombus bisect each other at right angles (90°). Let the intersection point be O:

• Half of diagonal \[BD = \frac{12}{2} = 6\text{ cm}\] (OB)
• Half of diagonal \[AC = \frac{16}{2} = 8\text{ cm}\] (AO)

In the right-angled triangle AOB, the hypotenuse AB is:

\[AB = \sqrt{AO^2 + OB^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10\text{ cm}\]

The angle x (∠CAB) is formed by diagonal AC and side AB. In right triangle AOB, \[\sin x = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{OB}{AB} = \frac{6}{10} = \frac{3}{5}\].

Therefore, \[\csc x = \frac{1}{\sin x} = \frac{5}{3} = 1\frac{2}{3}\].