Question

find the area of the rectangle with vertices h(-7, 4), i(-2, 5), j(0, -5), and k(-5, -6). write your answer as a whole number, decimal, or simplified radical. do not round. square units

Explanation:

Step1: Find length of one - side using distance formula

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 distance between $H(-7,4)$ and $I(-2,5)$.
$d_{HI}=\sqrt{(-2+7)^2+(5 - 4)^2}=\sqrt{5^2+1^2}=\sqrt{25 + 1}=\sqrt{26}$.

Step2: Find length of adjacent - side

Let's find the distance between $I(-2,5)$ and $J(0,-5)$.
$d_{IJ}=\sqrt{(0 + 2)^2+(-5 - 5)^2}=\sqrt{2^2+(-10)^2}=\sqrt{4 + 100}=\sqrt{104}=2\sqrt{26}$.

Step3: Calculate area of rectangle

The area of a rectangle $A$ is given by $A = l\times w$, where $l$ and $w$ are the lengths of two adjacent sides.
$A=d_{HI}\times d_{IJ}=\sqrt{26}\times2\sqrt{26}=2\times26 = 52$.

Answer:

52