Question

Three of the vertices of a parallelogram are A(2, 4), B(6,2) and C (8, 6).(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.

Answer 1

Given:

Three of the vertices of a parallelogram are given as

`\begin{gathered} A\left(2,4\right)_ \\ B\left(6,2\right) \\ C(8,6) \end{gathered}`

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.

Step-by-step explanation:

Take D coordinate as (x,y)

now midpoint of AC and BD is same so

`\begin{gathered} ((2+8)/(2),(4+6)/(2))=((x+6)/(2),(2+y)/(2)) \\ \\ (5,5)=((x+6)/(2),(2+y)/(2)) \\ \\ x=4,y=8 \end{gathered}`

midpoint of AC

`((2+8)/(2),(4+6)/(2))=(5,5)`

length of diagonal AC

`AC=√(36+4)=2√(10)`

perimeter of ABCD

`AB=√(16+4)=√(20)`
`BC=√(4+16)=√(20)`

perimeter is

`2(AB+BC)=4√(20)`

Final answer:

(b) Find the mid-point of diagonal AC

`\begin{equation*} (5,5) \end{equation*}`

(c) Find the fourth vertex D

`(4,8)`

(d) Find the length of diagonal AC

`2√(10)`

(e) Find the perimeter of ABCD.

`\begin{equation*} 4√(20) \end{equation*}`