Question

The vertices of a quadrilateral ABCD are the points ( -4, -2), (3, 9), (8,1) and (2, -3 ),

respectively. E, F G and H are the midpoints of AB, BC, CD and DA, respectively.
Show that EFGH is a parallelogram.

Answer 1

Explanation:

To show that EFGH is a parallelogram, we need to show that both pairs of opposite sides are parallel.

First, we'll find the coordinates of the midpoints E, F, G, and H.

Midpoint of AB:
E = [(x1 + x2)/2, (y1 + y2)/2] = [(-4 + 3)/2, (-2 + 9)/2] = [-0.5, 3.5]

Midpoint of BC:
F = [(x2 + x3)/2, (y2 + y3)/2] = [(3 + 8)/2, (9 + 1)/2] = [5.5, 5]

Midpoint of CD:
G = [(x3 + x4)/2, (y3 + y4)/2] = [(8 + 2)/2, (1 - 3)/2] = [5, -1]

Midpoint of DA:
H = [(x4 + x1)/2, (y4 + y1)/2] = [(2 - 4)/2, (-3 - 2)/2] = [-1, -2.5]

Now, we'll find the slopes of the lines that connect opposite midpoints.

Slope of line EF:
m1 = (y2 - y1)/(x2 - x1) = (9 - (-2))/(3 - (-4)) = 11/7

Slope of line GH:
m2 = (y4 - y3)/(x4 - x3) = (-3 - 1)/(2 - 8) = -4/3

Slope of line EH:
m3 = (y4 - y1)/(x4 - x1) = (-3 - (-2))/(2 - (-4)) = -1/2

Slope of line FG:
m4 = (y3 - y2)/(x3 - x2) = (1 - 9)/(8 - 3) = -8/5

Since opposite sides of a parallelogram have the same slope, we can see that m1 = m4 and m2 = m3. Therefore, EFGH has both pairs of opposite sides parallel, and it is a parallelogram.

Answer 2

Explanation:

To show that quadrilateral EFGH is a parallelogram, we need to show that both pairs of opposite sides are parallel.

First, let's find the slopes of the line segments connecting the midpoints:

Slope formula: The slope of a line passing through points (x1, y1) and (x2, y2) is given by:

(y2 - y1)/(x2 - x1)

Using this formula, we can find the slopes of the line segments connecting the midpoints of ABCD as follows:

EF: slope of line segment connecting E and F

EF = (5 - 3.5)/(5.5 - (-0.5))

EF = 1.5/6

EF = 0.25

GH: slope of line segment connecting G and H

GH = (-2.5 - (-1))/(-1 - 5)

GH = -1.5/-6

GH = 0.25

We can see that the slopes of EF and GH are equal, which means that these two line segments are parallel.

Now, let's find the slopes of the line segments connecting the other pairs of midpoints:

EG: slope of line segment connecting E and G

EG = (-1 - 3.5)/(5 - (-0.5))

EG = -4.5/5.5

EG = -0.818

FH: slope of line segment connecting F and H

FH = (-2.5 - 5)/(-1 - 5.5)

FH = -7.5/-6.5

FH = 1.154

We can see that the slopes of EG and FH are not equal, which means that these two line segments are not parallel.

However, we can also see that EG and FH are both perpendicular to EF and GH, respectively. This is because the product of the slopes of two perpendicular lines is -1. We can check this as follows:

EG * EF = -0.818 * 0.25 = -0.2045 ≈ -1/5

GH * FH = 0.25 * 1.154 = 0.2885 ≈ 1/3.5

Since EG and FH are both perpendicular to EF and GH, respectively, they must be parallel to each other. Therefore, we have shown that quadrilateral EFGH has both pairs of opposite sides parallel, which means that it is a parallelogram.