Give quadrilateral EFGH ,with coordinates E(1,4), F(1,2), G(5,0) and H(3,4); verify that EG bisects FH

Just verify that they intersect at (3,3)

To verify that EG bisects FH, we need to show that the midpoint of FH lies on the line segment EG.

Step 1: Find the coordinates of the midpoint of FH.
To find the midpoint of a line segment with endpoints F(1, 2) and H(3, 4), we use the midpoint formula:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

Substituting the coordinates of F and H:
Midpoint = ((1 + 3) / 2, (2 + 4) / 2)
Midpoint = (2, 3)

Therefore, the midpoint of FH is (2,3).

Step 2: Find the equation of the line EG.
To find the equation of the line EG, we need to determine the slope of EG and a point on the line.

Slope of EG = (change in y) / (change in x)
Slope of EG = (0 - 4) / (5 - 1)
Slope of EG = -4/4
Slope of EG = -1

Since we have a point E(1, 4), we can use the point-slope form to find the equation of EG:

y - y1 = m(x - x1)
y - 4 = -1(x - 1)
y - 4 = -x + 1
y = -x + 5

Therefore, the equation of EG is y = -x + 5.

Step 3: Verify that the midpoint of FH lies on EG.
To verify that the midpoint of FH lies on EG, we substitute the coordinates of the midpoint (2,3) into the equation of EG:

y = -x + 5
3 = -(2) + 5
3 = 3

Since both sides of the equation are equal, we can conclude that the midpoint (2,3) lies on EG.

Therefore, EG bisects FH in the given quadrilateral.