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State whether the statement is true or false.
The line segments of a quadrilateral joining the mid-points of the opposite sides bisect each other.

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True

False

answer is A.

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Detailed Solution

Given that, the line segments of a quadrilateral joining the mid-points of the opposite sides bisect each other.
Question Image

The points P, Q, R and S are the midpoints of AB, BC, CD and DA respectively. The diagonal is AC.
In

Δ A D C

, the midpoint of AD is S and CD is R. From midpoint theorem,
SR || AC and

S R = 12 A C

………… [1]
For

Δ A B C

, midpoint of AB is P and BC is Q. From midpoint theorem,
PQ || AC and

P Q = 12 A C

………... [2]
On comparing [1] and [2] we get:

P Q = S R = 12 A C and P Q | | S R

.
Similarly,

P S = Q R = 12 BD and PS | | Q R

.
A quadrilateral with a pair of equal parallel sides is a parallelogram. Thus, PQRS is a parallelogram.
The diagonals of a parallelogram bisect each other.

⇒ O S = O Q and O P = O R

Thus, the line segments joining the midpoints of the sides will bisect each other. Hence, the statement is true.
Therefore, the correct option is 1.

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