Question

Mikkel formed a flower garden and wants to place a fountain in the middle of the garden. Tocalculate the exact location, he placed a sketch of the garden in the coordinate plane. The fountainis to be placed where the diagonals of the quadrilateral intersect. What are the coordinates of thefountain? Enter your answer in the form (x,y) with no spaces.

Answer 1

To find the exact coordinates of the point where the diagonals intersect find the equation of each diagonal and then find the solution (where the lines cross each other)

Slope:

`m=(y_2-y_1)/(x_2-x_1)`

For the diagonal that passes for points D ( 0 , 0 ) and B ( 8 , 4)

Slope:

`m=(4-0)/(8-0)=(4)/(8)=(1)/(2)`

y-intrpcet (b):

`\begin{gathered} y=mx+b \\ 4=(1)/(2)(8)+b \\ 4=4+b \\ 4-4=b \\ 0=b \end{gathered}`

Equation:

`y=(1)/(2)x`

For the diagonal that passes for points A ( 1, 5) and C (6 , 2)

Slope:

`m=(2-5)/(6-1)=-(3)/(5)`

y-intercept (b):

`\begin{gathered} y=mx+b \\ 5=-(3)/(5)(1)+b \\ 5=-(3)/(5)+b \\ 5+(3)/(5)=b \\ (25+3)/(5)=b \\ \\ (28)/(5)=b \end{gathered}`

Equation:

`y=-(3)/(5)x+(28)/(5)`

-----------------------------

You have the next sytem of equations:

`\begin{gathered} y=(1)/(2)x \\ \\ y=-(3)/(5)x+(28)/(5) \end{gathered}`

To solve:

1. Substitute the value of y in the second equation for the one in the first equaion:

`(1)/(2)x=-(3)/(5)x+(28)/(5)`

2. Solve for x:

`\begin{gathered} (1)/(2)x+(3)/(5)x=(28)/(5) \\ \\ (5x+6x)/(10)=(28)/(5) \\ \\ (11)/(10)x=(28)/(5) \\ \\ x=((28)/(5))((10)/(11)) \\ \\ x=(280)/(55) \\ \\ x=(56)/(11) \end{gathered}`

3. Use the value of x to find the value of y:

`\begin{gathered} y=(1)/(2)x \\ \\ y=(1)/(2)((56)/(11)) \\ \\ y=(56)/(22) \\ \\ y=(28)/(11) \end{gathered}`Then, the point where the diagonas intersect is (56/11, 28/11) approximately (5.091,2.545)
`\begin{gathered} ((56)/(11),(28)/(11)) \\ \\ (5.091,2.545) \end{gathered}`