Question

the vertices of a quadrilateral ABCD are A(3,-2) B(0,-3) C(-2,3) D(4,1). what type of quadrilateral is ABCD?

Answer 1

Dear student
$$\begin{gathered} \mathrm{A}\left(3,-2\right),\mathrm{B}\left(0,-3\right),\mathrm{C}\left(-2,3\right),\mathrm{D}\left(4,1\right) \\ \mathrm{AB}=\sqrt{{\left(0-3\right)}^2+{\left(-3+2\right)}^2}=\sqrt{9+1}=\sqrt{10} \\ \mathrm{BC}=\sqrt{{\left(-2+0\right)}^2+{\left(3+3\right)}^2}=\sqrt{4+36}=\sqrt{40}=2\sqrt{10} \\ \mathrm{CD}=\sqrt{{\left(4+2\right)}^2+{\left(1-3\right)}^2}=\sqrt{36+4}=\sqrt{40}=2\sqrt{10} \\ \mathrm{AD}=\sqrt{{\left(3-4\right)}^2+{\left(-2-1\right)}^2}=\sqrt{1+9}=\sqrt{10} \\ \mathrm{Therefore},\mathrm{AB}=\mathrm{AD}\mathrm{and}\mathrm{BC}=\mathrm{CD} \\ \mathrm{Since}\mathrm{opposite}\mathrm{sides}\mathrm{are}\mathrm{equal}. \\ \mathrm{So},\mathrm{ABCD}\mathrm{is}\mathrm{a}\mathrm{rectangle}. \end{gathered}$$
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