Question

determine the coordinates of the vertices of the rectangle to compute the area of the rectangle using the distance formula (round to the nearest integer)
a 15 units²
b 30 units²
c 45 units²
d 50 units²

Explanation:

Step1: Assume vertices coordinates

Let the vertices of the rectangle be approximated as $(6,8)$, $(9,11)$, $(14,6)$ and $(11,3)$ (by visual - estimation from the graph).

Step2: Use distance formula for length and width

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Let's find the length $l$ and width $w$ of the rectangle.
For two adjacent vertices, say $(6,8)$ and $(9,11)$:
\[d_1=\sqrt{(9 - 6)^2+(11 - 8)^2}=\sqrt{9 + 9}=\sqrt{18}\approx 4.24\]
For another pair of adjacent vertices, say $(9,11)$ and $(14,6)$:
\[d_2=\sqrt{(14 - 9)^2+(6 - 11)^2}=\sqrt{25+25}=\sqrt{50}\approx7.07\]

Step3: Calculate the area

The area of a rectangle $A=l\times w$. So $A = d_1\times d_2\approx4.24\times7.07\approx30$

Answer:

B. 30 units²