Question

finding dimensions and area of a rectangle
a rectangular table is positioned in a 10 - foot by 10 - foot room as shown
how long is the longer edge of the table?
square root of feet
what is the area of the tabletop?
square feet

Explanation:

Step1: Identify two - point formula for distance

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Find the lengths of the sides of the rectangle

Let's assume two adjacent vertices of the rectangle are $(3,3)$ and $(6,8)$. Then the length of the side between them is $d_1=\sqrt{(6 - 3)^2+(8 - 3)^2}=\sqrt{3^2 + 5^2}=\sqrt{9+25}=\sqrt{34}$. Let another adjacent side has vertices $(6,8)$ and $(8,2)$. The length of this side is $d_2=\sqrt{(8 - 6)^2+(2 - 8)^2}=\sqrt{2^2+( - 6)^2}=\sqrt{4 + 36}=\sqrt{40}$. Since $\sqrt{40}>\sqrt{34}$, the longer edge is $\sqrt{40}$ feet.

Step3: Calculate the area of the rectangle

The area of a rectangle $A = l\times w$, where $l$ and $w$ are the lengths of the sides. Here $l=\sqrt{40}$ and $w=\sqrt{34}$, so $A=\sqrt{40}\times\sqrt{34}=\sqrt{40\times34}=\sqrt{1360}=2\sqrt{340}=4\sqrt{85}\approx 36.88$ square - feet.

Answer:

Square root of $40$ feet
$4\sqrt{85}$ square feet