Rhombus Area Calculation Details

This solution explains how to find the area of a rhombus when the side length and one diagonal are given.

Understanding Rhombus Properties

A rhombus is a quadrilateral with all four sides equal in length. Key properties include:

• All sides are equal (given as 13 cm).
• The diagonals bisect each other at right angles (90 degrees).
• The diagonals divide the rhombus into four congruent right-angled triangles.

Calculating the Second Diagonal

Let the side of the rhombus be '$s$', and the diagonals be '$d_1$' and '$d_2$'. We are given:

• Side, $s = 13$ cm
• One diagonal, $d_1 = 24$ cm

The diagonals of a rhombus bisect each other. This means they cut each other in half at their intersection point.

Consider one of the four right-angled triangles formed by the diagonals. The sides of this triangle are:

• Hypotenuse = side of the rhombus ($s = 13$ cm)
• One leg = half of the first diagonal ($d_1/2 = 24/2 = 12$ cm)
• Other leg = half of the second diagonal ($d_2/2$)

We can use the Pythagorean theorem ($a^2 + b^2 = c^2$) for this right-angled triangle:

$$ (d_1/2)^2 + (d_2/2)^2 = s^2 $$

Substitute the known values:

$$ (12)^2 + (d_2/2)^2 = (13)^2 $$

$$ 144 + (d_2/2)^2 = 169 $$

Now, solve for $(d_2/2)^2$:

$$ (d_2/2)^2 = 169 - 144 $$

$$ (d_2/2)^2 = 25 $$

Take the square root to find $d_2/2$:

$$ d_2/2 = \sqrt{25} $$

$$ d_2/2 = 5 \text{ cm} $$

Now, find the length of the second diagonal, $d_2$:

$$ d_2 = 2 \times 5 $$

$$ d_2 = 10 \text{ cm} $$

Calculating the Rhombus Area

The area of a rhombus can be calculated using the lengths of its diagonals with the formula:

$$ \text{Area} = \frac{1}{2} \times d_1 \times d_2 $$

Substitute the lengths of the diagonals ($d_1 = 24$ cm and $d_2 = 10$ cm):

$$ \text{Area} = \frac{1}{2} \times 24 \text{ cm} \times 10 \text{ cm} $$

$$ \text{Area} = 12 \text{ cm} \times 10 \text{ cm} $$

$$ \text{Area} = 120 \text{ cm}^2 $$

Conclusion

The area of the rhombus is 120 cm$^2$. This matches option 2.