Given:
Three of the vertices of a parallelogram are given as
[tex]\begin{gathered} A\left(2,4\right)_ \\ B\left(6,2\right) \\ C(8,6) \end{gathered}[/tex]Required:
(a) Plot the point A, B and C in the coordinate plane
(b) Find the mid-point of diagonal AC
(c) Find the fourth vertex D
(d) Find the length of diagonal AC
(e) Find the perimeter of ABCD.
Explanation:
Take D coordinate as (x,y)
now midpoint of AC and BD is same so
[tex]\begin{gathered} (\frac{2+8}{2},\frac{4+6}{2})=(\frac{x+6}{2},\frac{2+y}{2}) \\ \\ (5,5)=(\frac{x+6}{2},\frac{2+y}{2}) \\ \\ x=4,y=8 \end{gathered}[/tex]midpoint of AC
[tex](\frac{2+8}{2},\frac{4+6}{2})=(5,5)[/tex]length of diagonal AC
[tex]AC=\sqrt{36+4}=2\sqrt{10}[/tex]perimeter of ABCD
[tex]AB=\sqrt{16+4}=\sqrt{20}[/tex][tex]BC=\sqrt{4+16}=\sqrt{20}[/tex]perimeter is
[tex]2(AB+BC)=4\sqrt{20}[/tex]Final answer:
(b) Find the mid-point of diagonal AC
[tex]\begin{equation*} (5,5) \end{equation*}[/tex](c) Find the fourth vertex D
[tex](4,8)[/tex](d) Find the length of diagonal AC
[tex]2\sqrt{10}[/tex](e) Find the perimeter of ABCD.
[tex]\begin{equation*} 4\sqrt{20} \end{equation*}[/tex]
