To find the area of a rhombus and the length of its other diagonal given one of its diagonals, the side length, and the altitude, follow these steps:
1. Calculate the Area of the Rhombus:
The formula for the area (A) of a rhombus can be derived either from the product of its side length and altitude or from the product of its diagonals.
Given:
- Side length (s) = 5 cm
- Altitude (h) = 4.8 cm
The area can be calculated using:
[tex]\[
\text{Area} = \text{side} \times \text{altitude}
\][/tex]
Plugging in the given values:
[tex]\[
\text{Area} = 5 \, \text{cm} \times 4.8 \, \text{cm} = 24 \, \text{cm}^2
\][/tex]
2. Use the Area to Find the Length of the Other Diagonal:
The area of a rhombus can also be calculated using the formula involving its diagonals:
[tex]\[
\text{Area} = \frac{1}{2} \times d_1 \times d_2
\][/tex]
where [tex]\( d_1 \)[/tex] and [tex]\( d_2 \)[/tex] are the lengths of the diagonals.
We know the area and one of the diagonals:
[tex]\[
\text{Area} = 24 \, \text{cm}^2
\][/tex]
[tex]\[
d_1 = 8 \, \text{cm}
\][/tex]
Plugging these into the formula, we want to solve for [tex]\( d_2 \)[/tex]:
[tex]\[
24 = \frac{1}{2} \times 8 \times d_2
\][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[
48 = 8 \times d_2
\][/tex]
Now, solve for [tex]\( d_2 \)[/tex]:
[tex]\[
d_2 = \frac{48}{8} = 6 \, \text{cm}
\][/tex]
In conclusion:
- The area of the rhombus is [tex]\( 24 \, \text{cm}^2 \)[/tex].
- The length of the other diagonal is [tex]\( 6 \, \text{cm} \)[/tex].
