Question

what is the perimeter of rectangle efgh?
√10 + √29 units
2√10 + 2√29 units
22 units
39 units

Explanation:

Step1: Calculate the length of \(EH\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(E(1,-1)\) and \(H(2,2)\), \(x_1 = 1,y_1=-1,x_2 = 2,y_2 = 2\).
\(EH=\sqrt{(2 - 1)^2+(2+ 1)^2}=\sqrt{1 + 9}=\sqrt{10}\)

Step2: Calculate the length of \(GH\)

For \(G(-3,4)\) and \(H(2,2)\), \(x_1=-3,y_1 = 4,x_2 = 2,y_2 = 2\).
\(GH=\sqrt{(2 + 3)^2+(2 - 4)^2}=\sqrt{25+4}=\sqrt{29}\)

Step3: Use the perimeter formula of a rectangle

The perimeter of a rectangle \(P = 2(l + w)\), where \(l\) and \(w\) are the length and width of the rectangle.
Here \(l=\sqrt{29}\), \(w=\sqrt{10}\)
\(P=2(\sqrt{10}+\sqrt{29})=2\sqrt{10}+2\sqrt{29}\)

Answer:

\(2\sqrt{10}+2\sqrt{29}\) units (the second option)