Why does connecting the midpoints of any quadrilateral form a parallelogram?

The midpoints of the sides of an arbitrary quadrilateral form a parallelogram. If the quadrilateral is convex or concave (not complex), then the area of the parallelogram is half the area of the quadrilateral. The theorem can be generalized to the midpoint polygon of an arbitrary polygon.

How do you prove parallelogram?

There are five ways to prove that a quadrilateral is a parallelogram:

1. Prove that both pairs of opposite sides are congruent.
2. Prove that both pairs of opposite sides are parallel.
3. Prove that one pair of opposite sides is both congruent and parallel.
4. Prove that the diagonals of the quadrilateral bisect each other.

How do you prove Varignon’s Theorem?

Varignon’s Theorem: Moment of a force about any point is equal to the sum of the moments of the components of that force about the same point. which says that the moment of R about O equals the sum of the moments about O of its components P and Q . This proves the theorem.

How do you prove 4 points to make a parallelogram?

Let the points (4, 5) (7, 6) (4, 3) (1, 2) represent the points A, B, C and D. Opposite sides of the quadrilateral formed by the given four points are equal. Also the diagonals are unequal. Therefore, the given points form a parallelogram.

How do you know if a quadrilateral is always a parallelogram?

If you connect the midpoints of the sides of any quadrilateral, the resulting quadrilateral is always a parallelogram. Surprisingly, this is true whether it is a special kind of quadrilateral like a parallelogram or kite or trapezoid, or just any arbitrary simple convex quadrilateral with no parallel or equal sides.

How do you make a parallelogram from midpoints?

Parallelogram Formed by Connecting the Midpoints of a Quadrilateral 1 Problem. In a quadrilateral ABCD, the points P, Q, R and S are the midpoints of sides AB, BC, CD and DA, respectively. 2 Strategy. The fact that we are told that P, Q, R and S are the midpoints should remind us of the Triangle Midsegment… 3 Proof. More

How do you make a quadrilateral have opposite sides parallel?

When we connect the midpoints (the point exactly half-way along a line) of each side of the quadrilateral, one after the other, we create a new shape that has opposite sides parallel, even though the containing quadrilateral might not. To see why, click on the “Diags” button and drag the points around, while thinking “midpoints”.

How do you prove that the lines are parallel in geometry?

Theorem: If a transversal cuts across two lines and the alternate interior angles are congruent, then the lines are parallel The two-column proof proved the quadrilateral is a parallelogram by proving opposite sides were parallel. You can also use the paragraph proof form for any of the six ways.