Quadrilateral PQRS has vertices P(0,6), Q(-6,-2), R(2,-4), and S(4,2). Verify that the quadrilateral formed by joining the midpoints of the sides of PQRS is a parallelogram.

midpoints:

mPQ=A: (-3,2)
mQR=B: (-2,-3)
mRS=C: (3,-1)
mSP=D: (2,4)

check slopes:
AB: -5
BC: 2/5
CD: -5
DA: 2/5

parallel in pairs. OK.

Show tagt RSTU is a rectangle if the coordinates of tye vertices are respectively R(2,6) S(6,4) T(2,-4) and U(-2,-2)

To verify if the quadrilateral formed by joining the midpoints of the sides of PQRS is a parallelogram, we need to show that the opposite sides are equal and parallel.

Let's find the midpoints of each side first.

1. Midpoint of PQ:
Use the midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint of PQ = ((0 + (-6))/2, (6 + (-2))/2) = (-3, 2)

2. Midpoint of QR:
Midpoint of QR = ((-6 + 2)/2, (-2 + (-4))/2) = (-2, -3)

3. Midpoint of RS:
Midpoint of RS = ((2 + 4)/2, (-4 + 2)/2) = (3, -1)

4. Midpoint of SP:
Midpoint of SP = ((4 + 0)/2, (2 + 6)/2) = (2, 4)

Now, let's calculate the slopes of the opposite sides:

a. Slope of line PQ:
Slope of PQ = (y2 - y1) / (x2 - x1)
Slope of PQ = (-2 - 6) / (-6 - 0) = -8 / -6 = 4/3

b. Slope of line RS:
Slope of RS = (y2 - y1) / (x2 - x1)
Slope of RS = (2 - (-4)) / (4 - 2) = 6 / 2 = 3

c. Slope of line QR:
Slope of QR = (y2 - y1) / (x2 - x1)
Slope of QR = (-4 - (-2)) / (2 - (-6)) = (-2) / 8 = -1/4

d. Slope of line SP:
Slope of SP = (y2 - y1) / (x2 - x1)
Slope of SP = (6 - 2) / (0 - 4) = 4 / (-4) = -1

Now, let's compare the slopes:

a. Opposite sides PQ and RS:
The slope of PQ (4/3) is equal to the slope of RS (3), so they are parallel.

b. Opposite sides QR and SP:
The slope of QR (-1/4) is equal to the slope of SP (-1), so they are parallel.

Since the opposite sides are parallel, we have verified that the quadrilateral formed by joining the midpoints of the sides of PQRS is a parallelogram.

To verify that the quadrilateral formed by joining the midpoints of the sides of PQRS is a parallelogram, we need to show that both pairs of opposite sides are parallel.

Step 1: Find the midpoint of each side of the quadrilateral.
To find the midpoint of a segment with two endpoints (x₁, y₁) and (x₂, y₂), use the midpoint formula:

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

For the quadrilateral PQRS, let's calculate the midpoints of each side:

Midpoint of PQ:
x₁ = 0, y₁ = 6
x₂ = -6, y₂ = -2
Midpoint of PQ = ((0 + -6) / 2, (6 + -2) / 2) = (-3, 2)

Midpoint of QR:
x₁ = -6, y₁ = -2
x₂ = 2, y₂ = -4
Midpoint of QR = ((-6 + 2) / 2, (-2 + -4) / 2) = (-2, -3)

Midpoint of RS:
x₁ = 2, y₁ = -4
x₂ = 4, y₂ = 2
Midpoint of RS = ((2 + 4) / 2, (-4 + 2) / 2) = (3, -1)

Midpoint of SP:
x₁ = 4, y₁ = 2
x₂ = 0, y₂ = 6
Midpoint of SP = ((4 + 0) / 2, (2 + 6) / 2) = (2, 4)

Step 2: Calculate the slopes of the opposite sides.
To determine if two lines are parallel, we can examine their slopes. If the slopes are equal, the lines are parallel.

Slope = (y₂ - y₁) / (x₂ - x₁)

Using the midpoints we calculated, let's find the slopes of the opposite sides:

Slope of PQ and RS:
(x₁, y₁) = (-3, 2)
(x₂, y₂) = (3, -1)
Slope of PQ = (-1 - 2) / (3 - (-3)) = (-3) / 6 = -1/2

(x₁, y₁) = (3, -1)
(x₂, y₂) = (-3, 2)
Slope of RS = (2 - (-1)) / (-3 - 3) = 3 / (-6) = -1/2

Slope of QR and SP:
(x₁, y₁) = (-2, -3)
(x₂, y₂) = (2, 4)
Slope of QR = (4 - (-3)) / (2 - (-2)) = 7 / 4

(x₁, y₁) = (2, 4)
(x₂, y₂) = (-2, -3)
Slope of SP = (-3 - 4) / (-2 - 2) = (-7) / (-4) = 7/4

Step 3: Verify that the slopes are equal for both pairs of opposite sides.
Comparing the slopes, we see that:

Slope of PQ and RS = Slope of QR and SP = -1/2 = 7/4

Since both pairs of opposite sides have equal slopes, the quadrilateral formed by joining the midpoints of the sides of PQRS is a parallelogram.