Question

Calculate the perimeter and area of parallelogram QRST whose vertices are located at Q (-2, 3), R (6,3), S (3, -1) and T (-5, -1)

Answer 1

Area of QRST: 32
`un^2`.

Perimeter of QRST: 26 un.

Step-by-step explanation:

Before we solve for the area of the parallelogram, it's important to note that the formula for the area of a parallelogram is still base × height, the same formula as the area of a square or rectangle. This is because we can cut a parallelogram to form a rectangle with the same base length and vertical height. Since the base length of the parallelogram is 6 - (-2) = 6 + 2 = 8 units, and the vertical height of the parallelogram is 3 - (-1) = 3 + 1 = 4 units, that means the area of parallelogram QRST would be 8 × 4 = 32
`un^2`.

Since we already know the base length of parallelogram QRST - 8 units - and we need to find the perimeter of the parallelogram, we must find the length of the slant height as well. In order to do so, we can utilize the Pythagorean Theorem. Since the Pythagorean Theorem states that if you square the legs of a right triangle, find their sum, and then take the square root of the sum, you'll get the length of the hypotenuse of the right triangle.

In this scenario, we can set the slant height of parallelogram QRST as the hypotenuse, and set the change in x and y-values of the two points that form the slant height as the two legs of the right triangle. For example, if we use the two points Q and T as reference, their change in x is -2 - (-5) = -2 + 5 = 3 units, and their change in y is 3 - (-1) = 3 + 1 = 4 units. Therefore, the length of the slant height, or the hypotenuse, would be
`√(3^2+4^2) =√(9+16) =√(25) =5` units. Now that we know the length of the base and slant height, the perimeter of parallelogram QRST would be 8 × 2 + 5 × 2 = 16 + 10 = 26 units. Therefore, the perimeter of parallelogram QRST is 26 units long.

Have a great day! Feel free to let me know if you have any more questions :)

Answer 2

The area of parallelogram QRST is 32

Step-by-step explanation:

To find the perimeter of the parallelogram QRST, we need to add up the lengths of its four sides.

First, let's find the length of QR using the distance formula:

d(QR) = sqrt[(6 - (-2))^2 + (3 - 3)^2] = sqrt[64] = 8

Similarly, we can find the length of ST using the distance formula:

d(ST) = sqrt[(-5 - 3)^2 + (-1 - (-1))^2] = sqrt[64] = 8

To find the length of QS and TR, we can use the fact that they are parallel to the x-axis, so their lengths are simply the difference in their x-coordinates:

QS = 3 - (-2) = 5

TR = (-1) - 3 = -4

Now, we can add up the lengths of all four sides to get the perimeter:

perimeter = QR + QS + ST + TR = 8 + 5 + 8 + (-4) = 17

Therefore, the perimeter of parallelogram QRST is 17.

To find the area of the parallelogram, we can use the formula:

area = base x height

We can use the length of QR as the base, and the height can be found by dropping a perpendicular from point S to line QR.

The equation of the line passing through S and perpendicular to QR can be found by taking the negative reciprocal of the slope of QR, which is 0:

y - (-1) = -1/0 (x - 3)

x = 3

So, the height of the parallelogram is the distance between the line x = 3 and point S:

d(height) = 3 - (-1) = 4

Now we can use the formula for the area:

area = base x height = 8 x 4 = 32

Therefore, the area of parallelogram QRST is 32.