Given the coordinates of the vertices of a quadrilateral, determine whether it is a square, a rectangle or a
parallelogram. Then find the area of the quadrilateral.
E(–7, –1), F(2, –1), G(–4, –8), H(5, –8)
draw it
the top is at -1
the bottom is at - 8
so it is 7 somethings high (question written by mathematician and they do not do units)
the top runs from -7 to +2 so is 9 somethings long
the bottom runs from -4 to +5 so is also 9 somethings long
the top starts 3 somethings to the left of the bottom
clearly it is a parallelogram
area = height * base = 7 * 9 = 63 somethings^2
W = 2 - (-7) = 9.
L = 5 - (-4) = 9.
It is a square.
Area = 9 * 9 = 81 Square units.
left and right slope
To determine the type of quadrilateral and find its area, we need to follow a few steps:
1. Plot the given coordinates on a coordinate plane.
The coordinates of the vertices are:
E(–7, –1), F(2, –1), G(–4, –8), H(5, –8)
2. Connect the points to form the quadrilateral shape.
After plotting the points, connect them using straight lines to form the quadrilateral.
3. Determine the type of quadrilateral:
- If opposite sides are parallel and all angles are right angles, then it is a square.
- If opposite sides are parallel and all angles are not right angles, then it is a rectangle.
- If opposite sides are parallel, regardless of the angle measurements, then it is a parallelogram.
- If none of the above conditions are met, it is a general quadrilateral.
4. Measure the lengths of the sides:
To determine the type of quadrilateral and its area, we need to calculate the lengths of the sides.
- Calculate the length of each side by using the distance formula:
Side EF: distance between E and F
EF = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((2 - (-7))^2 + ((-1) - (-1))^2) = sqrt(81) = 9
Side FG: distance between F and G
FG = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((-4 - 2)^2 + ((-8) - (-1))^2) = sqrt(81) = 9
Side GH: distance between G and H
GH = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((5 - (-4))^2 + ((-8) - (-8))^2) = sqrt(81) = 9
Side HE: distance between H and E
HE = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((-7 - 5)^2 + ((-1) - (-8))^2) = sqrt(81) = 9
5. Determine the type of quadrilateral:
After calculating the lengths of the sides, we can see that all four sides are equal, EF = FG = GH = HE = 9.
Since all sides are equal, the quadrilateral is either a square or a rectangle.
Next, we need to examine the angles to differentiate between a square and a rectangle.
6. Measure the angles:
- Calculate the measurements of the angles. We can use the slope formula to find the slopes of the sides.
The formula for slope (m) is:
m = (y2 - y1) / (x2 - x1)
Slope EF: m = (-1 - (-1)) / (2 - (-7)) = 0/9 = 0
Slope FG: m = (-8 - (-1)) / (-4 - 2) = -7/-6
Slope GH: m = (-8 - (-8)) / (5 - (-4)) = 0/9 = 0
Slope HE: m = ((-1) - (-8)) / (-7 - 5) = 7/ (-12)
Now we have the slopes of each side. By comparing the slopes, we can determine the type of quadrilateral.
- If opposite sides have the same slope, then the quadrilateral could be a square or a rectangle.
- If only one pair of opposite sides have the same slope and the other pair of opposite sides have different slopes, then the quadrilateral is a parallelogram.
- If none of these conditions are met, it is a general quadrilateral.
In this case, we can see that opposite sides EF and GH have the same slope (0/9 = 0), and opposite sides FG and HE have different slopes (-7/-6 and 7/-12). Therefore, the quadrilateral is a parallelogram.
7. Calculate the area of the parallelogram:
To find the area of the parallelogram, we can use the formula:
Area = base * height
The base of the parallelogram can be EF or GH, and the height can be FG or HE. Since EF = GH = 9 (base) and FG = HE = 9 (height), we can choose any combination.
Area = base * height = 9 * 9 = 81 square units
Hence, the given quadrilateral is a parallelogram, and its area is 81 square units.