Question

find the perimeter of the polygon with the given vertices.

1. d(-3, 2), e(4, 2), f(4, -3)
2. g(-3, 2), h(2, 2), j(-1, -3)
3. k(-1, 1), l(4, 1), m(2, -2), n(-3, -2)
4. q(-4, -1), r(1, 4), s(4, 1), t(-1, -4)

find the area of the polygon with the given vertices.

1. g(2, 2), h(3, -1), j(-2, -1)
2. n(-1, 1), p(2, 1), q(2, -2), r(-1, -2)
3. f(-2, 3), g(1, 3), h(1, -1), j(-2, -1)
4. k(-3, 3), l(3, 3), m(3, -1), n(-3, -1)

Explanation:

Step1: Calculate side - lengths for perimeter

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), which simplifies for horizontal (\(y_1=y_2\)) and vertical (\(x_1=x_2\)) lines.

Step2: Sum side - lengths for perimeter

Add the lengths of all sides of the polygon to get the perimeter.

Step3: Identify base and height for area

For a triangle, find the base (a horizontal or vertical side) and the corresponding height (perpendicular distance from the opposite vertex to the base).

Step4: Calculate area

Use the formula \(A=\frac{1}{2}bh\) for the area of a triangle.

Answer:

1. For problem 3 (finding perimeter of polygon with vertices D(-3,2), E(4,2), F(4, - 3)):

• First, find the length of DE:
• Since D(-3,2) and E(4,2) have the same y - coordinate, the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) simplifies to \(d = |x_2 - x_1|\). Here, \(x_1=-3,x_2 = 4,y_1 = 2,y_2 = 2\), so \(DE=\sqrt{(4+3)^2+(2 - 2)^2}=7\).
• Second, find the length of EF:
• Since E(4,2) and F(4,-3) have the same x - coordinate, using the distance formula, \(EF=\sqrt{(4 - 4)^2+(-3 - 2)^2}=5\).
• Third, find the length of DF:
• Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) with \(x_1=-3,y_1 = 2,x_2 = 4,y_2=-3\), \(DF=\sqrt{(4 + 3)^2+(-3 - 2)^2}=\sqrt{49 + 25}=\sqrt{74}\approx8.6\).
• The perimeter \(P=DE + EF+DF=7 + 5+\sqrt{74}\approx7 + 5+8.6=20.6\) (There seems to be an error in the hand - written work where it just added DE and EF).

1. For problem 7 (finding area of polygon with vertices G(2,2), H(3,-1), J(-2,-1)):

• First, note that the base of the triangle formed by these points. Since H(3,-1) and J(-2,-1) have the same y - coordinate, the length of the base \(b=|3+2| = 5\).
• Second, find the height. The height is the vertical distance from the line \(y=-1\) to the point G(2,2), so \(h=|2+1| = 3\).
• Third, use the triangle area formula \(A=\frac{1}{2}bh\). Substituting \(b = 5\) and \(h = 3\), we get \(A=\frac{1}{2}\times5\times3 = 7.5\).