5 Kari makes a kite in the form of a parallelogram, EDGF, as shown in the diagram. Its diagonals [DF] and [EG] intersect at point H. She plots the vertices on a coordinate system with the following coordinates:
D(3,6), G(1,3), H(4, y) and E(x, 6). Find the values of x and y.
![5 Kari Makes A Kite In The Form Of A Parallelogram, EDGF, As Shown In The Diagram. Its Diagonals [DF]](https://brainimage.s3.us-east-005.backblazeb2.com/contents/attachments/swQQixrcLUxWzQXGlKVFqsNDt6ce90z5.png)
Answers
Answer:
The values of x and y are;
x = 7 and y = 4.5
Step-by-step explanation:
The question is a word problem, given that the shape of the kite Karl makes is a parallelogram
The coordinates of the vertices are;
D(3, 6), G(1, 3), H(4, y), and E(x, 6)
From the properties of a parallelogram we have;
The diagonals bisect each other
Therefore;
GH = EH = 1/2·GE
Therefore, we have;
The coordinates of the point H = The coordinate of the midpoint of the line EG
The coordinates of the midpoint of a line with end points, (x₁, y₁) and (x₂, y₂) is given as follows;
The coordinates of the midpoint of a line = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Therefore;
The coordinates of the midpoint of the line EG are ((1 + x)/2, (3 + 6)/2)
Given that the midpoint of EG is the point H and the coordinate of the point H is (4, y), we have;
(1 + x)/2 = 4, therefore;
2 × 4 = 1 + x
8 = 1 + x
x = 8 - 1 = 7
Similarly, we have from the coordinate of point H which is given as follows;
((1 + x)/2, (3 + 6)/2) = (4, y)
y = (3 + 6)/2 = 9/2 = 4.5
Therefore, x = 7, y = 4.5.
Parallel sides of a parallelogram have the same slope
The values of x and y are 6 and 3, respectively.
The coordinate points are given as:
[tex]\mathbf{D =(3,6)}[/tex]
[tex]\mathbf{G =(1,3)}[/tex]
[tex]\mathbf{H =(4,y)}[/tex]
[tex]\mathbf{E =(x,6)}[/tex]
Lines DG and HE are parallel.
This means that they have the same slope (m)
The slope is calculated as:
[tex]\mathbf{m = \frac{y_2 - y_1}{x_2 - x_1}}[/tex]
So, we have:
[tex]\mathbf{m_{DG} = \frac{3 - 6}{1 - 3}}[/tex]
[tex]\mathbf{m_{DG} = \frac{- 3}{- 2}}[/tex]
[tex]\mathbf{m_{DG} = \frac{3}{2}}[/tex]
Similarly
[tex]\mathbf{m_{HE} = \frac{6 - y}{x - 4}}[/tex]
The slopes are equal.
So, we have:
[tex]\mathbf{\frac{6 - y}{x - 4} = \frac 32}[/tex]
By comparison:
[tex]\mathbf{6 - y = 3}[/tex] and [tex]\mathbf{x - 4 = 2}[/tex]
Solve for y in [tex]\mathbf{6 - y = 3}[/tex]
[tex]\mathbf{y = 3}[/tex]
Solve for x in [tex]\mathbf{x - 4 = 2}[/tex]
[tex]\mathbf{x = 6}[/tex]
Hence, the values of x and y are 6 and 3, respectively.
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Hope this helped
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Hey there!☺
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