The area of a rhombus with side 13 cm and one diagonal 10 cm will be :

Question

Accepted Answer

To find the area of a rhombus when one side and one diagonal are given, we can use the properties of a rhombus and the Pythagorean theorem.

Given:
- Side of the rhombus = 13 cm
- One diagonal (let's say $d_1$) = 10 cm

In a rhombus, the diagonals bisect each other at right angles. Therefore, each half of the diagonal $d_1$ is $5 \ \text{cm}$.

Let's denote the other diagonal as $d_2$. Since the diagonals bisect each other, each half of $d_2$ will be $\frac{d_2}{2}$.

Using the Pythagorean theorem in one of the right triangles formed by the diagonals, we have:
$$
\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = \text{side}^2
$$
$$
5^2 + \left(\frac{d_2}{2}\right)^2 = 13^2
$$
$$
25 + \left(\frac{d_2}{2}\right)^2 = 169
$$
$$
\left(\frac{d_2}{2}\right)^2 = 169 - 25
$$
$$
\left(\frac{d_2}{2}\right)^2 = 144
$$
$$
\frac{d_2}{2} = 12
$$
$$
d_2 = 24 \ \text{cm}
$$

Now, the area of a rhombus is given by the formula:
$$
\text{Area} = \frac{1}{2} \times d_1 \times d_2
$$
$$
\text{Area} = \frac{1}{2} \times 10 \times 24
$$
$$
\text{Area} = 120 \ \text{cm}^2
$$

Therefore, the area of the rhombus is $120 \ \text{cm}^2$, which corresponds to Option 4.

Explanation for why other options are incorrect:
1. Option 1: $100 \ \text{cm}^2$ - This value is incorrect because the calculation using the diagonals does not yield this area.
2. Option 2: $105 \ \text{cm}^2$ - This value is incorrect as it does not match the calculated area using the given side and diagonal.
3. Option 3: $110 \ \text{cm}^2$ - This value is incorrect because the correct calculation results in a different area.

Thus, the correct answer is indeed Option 4, $120 \ \text{cm}^2$.