How do you find the midpoint of each diagonal of the quadrilateral with vertices P(1,3), Q(6,5), R(8,0), and S(3,2)?
To find the midpoint of each diagonal of the quadrilateral with vertices P(1,3), Q(6,5), R(8,0), and S(3,2), we can use the midpoint formula.
The midpoint formula states that the midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by the coordinates:
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
For the first diagonal, we need to find the midpoint of the line segment connecting points P(1,3) and R(8,0).
Using the midpoint formula, we have:
Midpoint₁ = ((1 + 8)/2, (3 + 0)/2) = (9/2, 3/2)
For the second diagonal, we need to find the midpoint of the line segment connecting points Q(6,5) and S(3,2).
Using the midpoint formula, we have:
Midpoint₂ = ((6 + 3)/2, (5 + (2))/2) = (9/2, 3/2)
Therefore, the midpoints of the diagonals of the quadrilateral are Midpoint₁ = (9/2, 3/2) and Midpoint₂ = (9/2, 3/2).
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Midpoint of the diagonal PR :
Midpoint of the diagonal QS :
Both the diagonals have the same Midpoint, and we have a Parallelogram.
We are given a Quadrilateral with the following Vertices:
The MidPoint Formula for a Line Segment with Vertices
Consider the Vertices
Using the Midpoint formula we can write
Consider the Vertices
Using the Midpoint formula we can write
By observing the two Intermediate results 1 and 2, we understand that both the diagonals have the same Midpoint, and hence the given Quadrilateral with four vertices is a Parallelogram.
Please refer to the image of the graph constructed using GeoGebra given below:
MPPR
MPQS
Some interesting properties of a parallelogram to remember:

Opposite sides of a parallelogram have the same length and hence they are congruent.

Opposite angles of the parallelogram have the same size/measure.

Obviously, opposite sides of a parallelogram are also parallel.

The diagonals of a parallelogram bisect each other.

Each diagonal of a parallelogram separates it into two congruent triangles.

We observe that our parallelogram has all sides congruent, and hence our parallelogram is a rhombus.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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