445,872 views
33 votes
33 votes
Heya!


\underline{ \underline{ \text{QUESTION : }}} In the given quadrilateral PQRS, the mid-points of the sides PQ , QR , RS and SP are A , B , C and D respectively. Prove that ABCD is a parallelogram.

Heya! \underline{ \underline{ \text{QUESTION : }}} In the given quadrilateral PQRS-example-1
User Schalk Versteeg
by
2.7k points

2 Answers

20 votes
20 votes

Answer:

this is your answer look it once

Heya! \underline{ \underline{ \text{QUESTION : }}} In the given quadrilateral PQRS-example-1
User Addedlovely
by
2.6k points
18 votes
18 votes

Answer:

See Below.

Explanation:

We are given that A, B, C, and D are the midpoints of sides PQ, QR, RS, and SP, respectively.

And we want to prove that ABCD is a parallelogram.

By the definition of midpoint, this means that:


SD\cong DP, \, PA\cong AQ, \, QB\congBR, \, \text{ and } RC\cong CS

To prove, we can construct a segment from S to Q to form SQ. This is shown in the first diagram.

By the Midpoint Theorem:


DA\parallel SQ

Similarly:


CB\parallel SQ

By the transitive property for parallel lines:


DA\parallel CB

Likewise, we can do the same for the other pair of sides. We will construct a segment from P to R to form PR. This is shown in the second diagram.

By the Midpoint Theorem:


AB\parallel PR

Similarly:


DC\parallel PR

So:


AB\parallel DC

This yields:


DA\parallel CB\text{ and } DC\parallel AB

By the definition of a parallelogram, it follows that:


ABCD\text{ is a parallelogram.}

Heya! \underline{ \underline{ \text{QUESTION : }}} In the given quadrilateral PQRS-example-1
Heya! \underline{ \underline{ \text{QUESTION : }}} In the given quadrilateral PQRS-example-2
User RomanS
by
2.4k points