Question

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 \\(\square\\) feet what is the area of the tabletop? \\(\square\\) square feet

Explanation:

Step1: Identify table vertices

Vertices: $(2,3)$, $(5,2)$, $(7,8)$, $(4,9)$

Step2: Calculate side lengths (distance formula)

First side (between $(2,3)$ & $(5,2)$):
$\sqrt{(5-2)^2 + (2-3)^2} = \sqrt{3^2 + (-1)^2} = \sqrt{9+1} = \sqrt{10}$
Second side (between $(5,2)$ & $(7,8)$):
$\sqrt{(7-5)^2 + (8-2)^2} = \sqrt{2^2 + 6^2} = \sqrt{4+36} = \sqrt{40}$

Step3: Confirm longer edge value

$\sqrt{40}$ is longer, so the radicand is 40.

Step4: Calculate table area

Area of rectangle = product of side lengths
$\sqrt{10} \times \sqrt{40} = \sqrt{10 \times 40} = \sqrt{400} = 20$

Answer:

Longer edge: $\boldsymbol{40}$ (inside the square root blank)
Area: $\boldsymbol{20}$ square feet