Question

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral is a​

Answer 1

GIVEN:

A quadrilateral PQRS of any type

To Find :

The type of  quadrilateral ABCD formed by joining the mid-points of the sides of the given quadrilateral .

Construction:

Let us join PR

Solution:

Referring the attached figure

In The Δ PQR

$\because$  A and B are mid points of sides PQ and QR

$\therefore$  By the mid point theorem

$AB \parallel PQ-------------(1)\\\\and\,\, AB =\frac{PQ}{2}-----------(2)\\\\Similarly \, in \, the\,\triangle PRS \\\\CD \parallel PQ-----------------(3)\\\\and \,\, CD =\frac{PQ}{2}.--------------(4)\\\\\\\ From\, (1) \,and\,(3)\, we\, get\\\\AB\parallel CD\\\\ From \,(2) \,and\,(4)\, we get\\\\AB=CD$

As per the definition of parallelogram we know

"If the opposite sides of a quadrilateral are parallel and equal

then it is a Parallelogram"

Thus ABCD is a parallelogram

Result:

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral is a​ parallelogram

Concepts used:

1. Mid point theorem
2. Definition of a parallelogram

Other useful type of parallelogram:

1. Any parallelogram with all sides equal but different angle is a Rhombus
2. Any parallelogram with all sides and all angles equal is a square
3. The parallelogram with opposite sides equal and all angles 90° is a rectangle
4. If two vectors act along two adjacent sides of a parallelogram then the  resultant vector is represented by the diagonal passing through the intersection of two vectors.

Answer 2

Answer:

Parallelogram

Step-by-step explanation:

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral is a​ Parallelogram.

Some important points about Parallelogram:

♦ In a parallelogram,  opposite angles and opposite sides are equal are equal and diagonals bisect each other.

♦ Rectangle and Square are also the examples of Parallelogram, whose opposite sides are equal and each angle is 90.

♦ Adjacent angles are supplementary.

♦ Each diagonal divides the parallelogram into two congruent triangles.