Question

13 multiple choice 1 point given rectangle abcd with vertices a(1, 4), b(3, 6), c(6, 3), and d(4, 1), what is the area of abcd? leave your answer in simplified radical form. 12 square units 10 square units 6√2 square units 5√2 square units

Explanation:

Step1: Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ to find the length of two adjacent sides. First, find the length of side AB with $A(1,4)$ and $B(3,6)$.

$AB=\sqrt{(3 - 1)^2+(6 - 4)^2}=\sqrt{2^2+2^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}$

Step2: Then, find the length of side BC with $B(3,6)$ and $C(6,3)$.

$BC=\sqrt{(6 - 3)^2+(3 - 6)^2}=\sqrt{3^2+( - 3)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$

Step3: Since the area of a rectangle $A = l\times w$, where $l$ and $w$ are the lengths of adjacent sides.

$A=AB\times BC=2\sqrt{2}\times3\sqrt{2}=2\times3\times\sqrt{2}\times\sqrt{2}=6\times2 = 12$

Answer:

12 square units