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.

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:

Lines DG and HE are parallel.

This means that they have the same slope (m)

The slope is calculated as:

So, we have:

Similarly

The slopes are equal.

So, we have:

By comparison:

and

Solve for y in

Solve for x in

Hence, the values of x and y are 6 and 3, respectively.

Read more about parallelogram and slopes at:

link

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.