Answer:
(A)Segment EF, segment FG, segment GH, and segment EH are congruent
Step-by-step explanation:
<u>Step 1</u>
Quadrilateral EFGH with points E(-2,3), F(1,6), G(4,3), H(1,0)
<u>Step 2</u>
Using the distance formula
![Distance=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}](https://tex.z-dn.net/?f=Distance%3D%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D)
Given E(-2,3), F(1,6)
![|EF|=\sqrt{(6-3)^2+(1-(-2))^2}=\sqrt{3^2+3^2}=\sqrt{18}=3\sqrt{2}](https://tex.z-dn.net/?f=%7CEF%7C%3D%5Csqrt%7B%286-3%29%5E2%2B%281-%28-2%29%29%5E2%7D%3D%5Csqrt%7B3%5E2%2B3%5E2%7D%3D%5Csqrt%7B18%7D%3D3%5Csqrt%7B2%7D)
Given F(1,6), G(4,3)
![|FG|=\sqrt{(3-6)^2+(4-1)^2}=\sqrt{3^2+3^2}=\sqrt{18}=3\sqrt{2}](https://tex.z-dn.net/?f=%7CFG%7C%3D%5Csqrt%7B%283-6%29%5E2%2B%284-1%29%5E2%7D%3D%5Csqrt%7B3%5E2%2B3%5E2%7D%3D%5Csqrt%7B18%7D%3D3%5Csqrt%7B2%7D)
Given G(4,3), H(1,0)
![|GH|=\sqrt{(0-3)^2+(1-4)^2}=\sqrt{(-3)^2+(-3)^2}=\sqrt{18}=3\sqrt{2}](https://tex.z-dn.net/?f=%7CGH%7C%3D%5Csqrt%7B%280-3%29%5E2%2B%281-4%29%5E2%7D%3D%5Csqrt%7B%28-3%29%5E2%2B%28-3%29%5E2%7D%3D%5Csqrt%7B18%7D%3D3%5Csqrt%7B2%7D)
Given E (−2, 3), H (1, 0)
![|EH|=\sqrt{(0-3)^2+(1-(-2))^2}=\sqrt{(-3)^2+(3)^2}=\sqrt{18}=3\sqrt{2}](https://tex.z-dn.net/?f=%7CEH%7C%3D%5Csqrt%7B%280-3%29%5E2%2B%281-%28-2%29%29%5E2%7D%3D%5Csqrt%7B%28-3%29%5E2%2B%283%29%5E2%7D%3D%5Csqrt%7B18%7D%3D3%5Csqrt%7B2%7D)
<u>Step 3</u>
Segment EF ,E (−2, 3), F (1, 6)
Slope of |EF|=![\frac{6-3}{1+2} =\frac{3}{3}=1](https://tex.z-dn.net/?f=%5Cfrac%7B6-3%7D%7B1%2B2%7D%20%3D%5Cfrac%7B3%7D%7B3%7D%3D1)
Segment GH, G (4, 3), H (1, 0)
Slope of |GH|![= \frac{0-3}{1-4} =\frac{-3}{-3}=1](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B0-3%7D%7B1-4%7D%20%3D%5Cfrac%7B-3%7D%7B-3%7D%3D1)
<u>Step 4</u>
Segment EH, E(−2, 3), H (1, 0)
Slope of |EH|![= \frac{0-3}{1+2} =\frac{-3}{3}=-1](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B0-3%7D%7B1%2B2%7D%20%3D%5Cfrac%7B-3%7D%7B3%7D%3D-1)
Segment FG, F (1, 6,) G (4, 3)
Slope of |EH| ![=\frac{3-6}{4-1} =\frac{-3}{3}=-1](https://tex.z-dn.net/?f=%3D%5Cfrac%7B3-6%7D%7B4-1%7D%20%3D%5Cfrac%7B-3%7D%7B3%7D%3D-1)
<u>Step 5</u>
Segment EF and segment GH are perpendicular to segment FG.
The slope of segment EF and segment GH is 1. The slope of segment FG is −1.
<u>Step 6</u>
<u>Segment EF, segment FG, segment GH, and segment EH are congruent. </u>
The slope of segment FG and segment EH is −1. The slope of segment GH is 1.
<u>Step 7</u>
All sides are congruent, opposite sides are parallel, and adjacent sides are perpendicular. Quadrilateral EFGH is a square