Answer :
Since the park is in the shape of a parallelogram, and we know the lengths of two adjacent sides and one diagonal, we can find the area of the park by finding the area of the triangle formed by the two sides and the diagonal and doubling it, as each parallelogram can be divided into two congruent triangles by one of its diagonals.
Here are the steps to calculate the area of the triangle using Heron's formula:
1. Calculate the semi-perimeter of the triangle, which is half the sum of all three sides. Let a = 51m, b = 52m, and c = 53m (the diagonal):
[tex]\( s = \frac{a + b + c}{2} = \frac{51 + 52 + 53}{2} = \frac{156}{2} = 78 \)[/tex] m
2. Now, use Heron's formula to find the area of the triangle:
[tex]\( A_{\text{triangle}} = \sqrt{s(s - a)(s - b)(s - c)} \)[/tex]
[tex]\( A_{\text{triangle}} = \sqrt{78(78 - 51)(78 - 52)(78 - 53)} \)[/tex]
[tex]\( A_{\text{triangle}} = \sqrt{78(27)(26)(25)} \)[/tex]
[tex]\( A_{\text{triangle}} = \sqrt{78272625} \)[/tex]
3. We have the values for calculation:
[tex]\( A_{\text{triangle}} = \sqrt{78272625} \)[/tex]
First calculate the product inside the square root:
[tex]\( A_{\text{triangle}} = \sqrt{5463900} \)[/tex]
4. Now calculate the square root of that product:
[tex]\( A_{\text{triangle}} = 2337 \)[/tex] (NOTE: The exact square root would be a decimal, but for the purpose of the provided options, we're using this number as an estimation)
5. The area of the parallelogram is twice the area of the triangle:
[tex]\( A_{\text{park}} = 2 \times A_{\text{triangle}} \)[/tex]
[tex]\( A_{\text{park}} = 2 \times 2337 \)[/tex]
[tex]\( A_{\text{park}} = 4674 \)[/tex] m²
The area of the park (the parallelogram) is therefore 4674 square meters. Since the options you've provided give incomplete information (i.e., "(iv) 1 33" and "(v)"), I cannot match this result to any of the options. Please double-check the options provided for completeness.
Here are the steps to calculate the area of the triangle using Heron's formula:
1. Calculate the semi-perimeter of the triangle, which is half the sum of all three sides. Let a = 51m, b = 52m, and c = 53m (the diagonal):
[tex]\( s = \frac{a + b + c}{2} = \frac{51 + 52 + 53}{2} = \frac{156}{2} = 78 \)[/tex] m
2. Now, use Heron's formula to find the area of the triangle:
[tex]\( A_{\text{triangle}} = \sqrt{s(s - a)(s - b)(s - c)} \)[/tex]
[tex]\( A_{\text{triangle}} = \sqrt{78(78 - 51)(78 - 52)(78 - 53)} \)[/tex]
[tex]\( A_{\text{triangle}} = \sqrt{78(27)(26)(25)} \)[/tex]
[tex]\( A_{\text{triangle}} = \sqrt{78272625} \)[/tex]
3. We have the values for calculation:
[tex]\( A_{\text{triangle}} = \sqrt{78272625} \)[/tex]
First calculate the product inside the square root:
[tex]\( A_{\text{triangle}} = \sqrt{5463900} \)[/tex]
4. Now calculate the square root of that product:
[tex]\( A_{\text{triangle}} = 2337 \)[/tex] (NOTE: The exact square root would be a decimal, but for the purpose of the provided options, we're using this number as an estimation)
5. The area of the parallelogram is twice the area of the triangle:
[tex]\( A_{\text{park}} = 2 \times A_{\text{triangle}} \)[/tex]
[tex]\( A_{\text{park}} = 2 \times 2337 \)[/tex]
[tex]\( A_{\text{park}} = 4674 \)[/tex] m²
The area of the park (the parallelogram) is therefore 4674 square meters. Since the options you've provided give incomplete information (i.e., "(iv) 1 33" and "(v)"), I cannot match this result to any of the options. Please double-check the options provided for completeness.
