Answer :
To find the area of rectangle [tex]\(DEFG\)[/tex] with the given vertices [tex]\(D(-8, 2)\)[/tex], [tex]\(E(2, 7)\)[/tex], [tex]\(F(5, 1)\)[/tex], and [tex]\(G(-5, -4)\)[/tex], we need to follow these steps:
1. Calculate the Length of Side [tex]\(DE\)[/tex]:
- Use the distance formula to find the length between points [tex]\(D\)[/tex] and [tex]\(E\)[/tex].
[tex]\[
\text{Length of } DE = \sqrt{(E_x - D_x)^2 + (E_y - D_y)^2} = \sqrt{(2 - (-8))^2 + (7 - 2)^2}
\][/tex]
[tex]\[
= \sqrt{(2 + 8)^2 + (5)^2} = \sqrt{10^2 + 5^2} = \sqrt{100 + 25} = \sqrt{125} = 11.18
\][/tex]
2. Calculate the Length of Side [tex]\(EF\)[/tex]:
- Again, use the distance formula for points [tex]\(E\)[/tex] and [tex]\(F\)[/tex].
[tex]\[
\text{Length of } EF = \sqrt{(F_x - E_x)^2 + (F_y - E_y)^2} = \sqrt{(5 - 2)^2 + (1 - 7)^2}
\][/tex]
[tex]\[
= \sqrt{(3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} = 6.71
\][/tex]
3. Calculate the Area of the Rectangle:
- The area of a rectangle is given by multiplying the lengths of two adjacent sides.
[tex]\[
\text{Area} = \text{Length of } DE \times \text{Length of } EF = 11.18 \times 6.71 = 75
\][/tex]
Therefore, the area of rectangle [tex]\(DEFG\)[/tex] is [tex]\(75\)[/tex] square units.
The correct answer is [tex]\(B. \, 75 \, u^2\)[/tex].
1. Calculate the Length of Side [tex]\(DE\)[/tex]:
- Use the distance formula to find the length between points [tex]\(D\)[/tex] and [tex]\(E\)[/tex].
[tex]\[
\text{Length of } DE = \sqrt{(E_x - D_x)^2 + (E_y - D_y)^2} = \sqrt{(2 - (-8))^2 + (7 - 2)^2}
\][/tex]
[tex]\[
= \sqrt{(2 + 8)^2 + (5)^2} = \sqrt{10^2 + 5^2} = \sqrt{100 + 25} = \sqrt{125} = 11.18
\][/tex]
2. Calculate the Length of Side [tex]\(EF\)[/tex]:
- Again, use the distance formula for points [tex]\(E\)[/tex] and [tex]\(F\)[/tex].
[tex]\[
\text{Length of } EF = \sqrt{(F_x - E_x)^2 + (F_y - E_y)^2} = \sqrt{(5 - 2)^2 + (1 - 7)^2}
\][/tex]
[tex]\[
= \sqrt{(3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} = 6.71
\][/tex]
3. Calculate the Area of the Rectangle:
- The area of a rectangle is given by multiplying the lengths of two adjacent sides.
[tex]\[
\text{Area} = \text{Length of } DE \times \text{Length of } EF = 11.18 \times 6.71 = 75
\][/tex]
Therefore, the area of rectangle [tex]\(DEFG\)[/tex] is [tex]\(75\)[/tex] square units.
The correct answer is [tex]\(B. \, 75 \, u^2\)[/tex].
