To find the area of the rhombus given the coordinates of three consecutive vertices [tex]\((2,-3)\)[/tex], [tex]\((6,5)\)[/tex], and [tex]\((-2,1)\)[/tex], we will follow these steps:
1. Determine the lengths of the diagonals:
- Let's denote the points as [tex]\(A(2, -3)\)[/tex], [tex]\(B(6, 5)\)[/tex], and [tex]\(C(-2, 1)\)[/tex].
- Calculate the length of diagonal [tex]\(AC\)[/tex] (linking [tex]\(A\)[/tex] and [tex]\(C\)[/tex]), and [tex]\(AB\)[/tex] (the distance from [tex]\(A\)[/tex] to [tex]\(B\)[/tex]).
The distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Calculate [tex]\(AC\)[/tex]:
[tex]\[
AC = \sqrt{(-2 - 2)^2 + (1 + 3)^2} = \sqrt{(-4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\][/tex]
Numerically, [tex]\( \sqrt{32} \approx 5.656854249 \)[/tex].
Calculate [tex]\(AB\)[/tex]:
[tex]\[
AB = \sqrt{(6 - 2)^2 + (5 + 3)^2} = \sqrt{(4)^2 + (8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}
\][/tex]
Numerically, [tex]\( \sqrt{80} \approx 8.944271910 \)[/tex].
2. Calculate the area of the rhombus:
- The area [tex]\(A\)[/tex] of a rhombus can be determined using the lengths of its diagonals:
[tex]\[
A = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2
\][/tex]
Substituting in the values we calculated for the diagonals:
[tex]\[
A = \frac{1}{2} \times 4\sqrt{2} \times 4\sqrt{5}
\][/tex]
Simplifying:
[tex]\[
A = \frac{1}{2} \times 8\sqrt{10}
\][/tex]
Numerically:
[tex]\[
A \approx \frac{1}{2} \times 4\sqrt{2} \times 4\sqrt{5} \approx \frac{1}{2} \times 5.656854249 \times 8.944271910 \approx 25.2982 \text{ (approximately)}
\][/tex]
3. Determine the closest area value from the given options:
- The given options are 24, 36, 18, and 48.
- The numerically computed area is approximately [tex]\(25.2982\)[/tex].
Comparing this with the provided choices, the closest value to [tex]\(25.2982\)[/tex] is [tex]\(24\)[/tex].
Hence, the area of the rhombus is closest to [tex]\(24\)[/tex], corresponding to option:
A) 24.
