ABCD is a quadrilateral in which p , q, r and s are mid points of side ab bc cd da. ac is a diagonal show that
1. SR paralell to ac and SR =1/2 ac
2. PQ =RS
3. PQRS is a parallelogram
Answers
Answer:
The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.
(i) In △DAC , S is the mid point of DA and R is the mid point of DC. Therefore, SR∥AC and SR=
2
1
AC.By mid-point theorem.
(ii) In △BAC , P is the mid point of AB and Q is the mid point of BC. Therefore, PQ∥AC and PQ=
2
1
AC.By mid-point theorem. But from (i) SR=
2
1
AC therefore PQ=SR
(iii) PQ∥AC & SR∥AC therefore PQ∥SR and PQ=SR. Hence, a quadrilateral with opposite sides equal and paralle is a parallelogram. Therefore PQRS is a parallelogram.
Answer
The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.
(i) In △DAC , S is the mid point of DA and R is the mid point of DC. Therefore, SR∥AC and SR= 1/ 2
AC.By mid-point theorem.
(ii) In △BAC , P is the mid point of AB and Q is the mid point of BC. Therefore, PQ∥AC and PQ=
1/2
AC.By mid-point theorem. But from (i) SR=
1 /2
AC therefore PQ=SR
(iii) PQ∥AC & SR∥AC therefore PQ∥SR and PQ=SR. Hence, a quadrilateral with opposite sides equal and parallel is a parallelogram. Therefore PQRS is a parallelogram.