Full solution

Q. given that coordinates of the following are

P(-1,-2)

Q(1,4)

R(3,0)

S(\frac{7}{3},-2)

Calculate area of trapezium

Identify Parallel Sides: To find the area of a trapezium, we need to identify the parallel sides and the distance (height) between them. Points $Q(1,4)$ and $R(3,0)$ seem to form one side, and points $P(-1,-2)$ and $S(\frac{7}{3},-2)$ seem to form the other parallel side since they have the same $y$ -coordinates.
Calculate Length of QR: Calculate the length of QR using the distance formula: $d = \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}$ . $\newline$ Length of QR = $\sqrt{((3 - 1)^2 + (0 - 4)^2)} = \sqrt{(4 + 16)} = \sqrt{20}$ .
Calculate Length of PS: Calculate the length of PS using the distance formula: $d = \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}$ . $\newline$ Length of PS = $\sqrt{\left(\frac{7}{3} - (-1)\right)^2 + (-2 - (-2))^2} = \sqrt{\left(\frac{10}{3}\right)^2 + 0} = \sqrt{\frac{100}{9}} = \frac{10}{3}$ .
Calculate Height: The height $h$ of the trapezium is the distance between the parallel sides QR and PS. We can use the y-coordinates of Q and P or R and S for this since they lie on the same vertical lines. $\newline$ Height $h$ = $|y_Q - y_P| = |4 - (-2)| = 6$ .
Calculate Area: Now, we can calculate the area of the trapezium using the formula: Area = $\frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$ . $\newline$ Area = $\frac{1}{2} \times (QR + PS) \times h = \frac{1}{2} \times (\sqrt{20} + \frac{10}{3}) \times 6$ .
Perform Calculation: Perform the calculation: $\text{Area} = \frac{1}{2} \times (\sqrt{20} + \frac{10}{3}) \times 6 = \frac{1}{2} \times (4.472 + 3.333) \times 6 = \frac{1}{2} \times 7.805 \times 6$ .
Final Calculation: Final calculation: $\text{Area} = \frac{1}{2} \times 7.805 \times 6 = 23.415 \times 3 = 70.245$ .