Question

1. ABCD is a quadrilateral in which P, Q, R and S are
mid-points of the sides AB, BC, CD and DA
(see Fig 8.29). AC is a diagonal. Show that:
D
R
(1) SRII AC and SR=
1 / 1
AC
S
(ii) PQ=SR
(ii) PQRS is a parallelogram.
A
B
P​

Answer 1

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.

Step-by-step explanation:

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Answer 2

Answer:

1) In ∆ACD,

S is the midpoint of AD and R is the midpoint of CD. SR=1/2AC & SR||AC...(1)

( by mid point theorem)

2) In ∆ABC,

P is the midpoint of AB and Q is the midpoint of BC. PQ=1/2AC & PQ||AC...(2)

(by mid point theorem)

from (1)&(2)

PQ=1/2 AC=SR and PQ||AC||SR

-: PQ=SR and PQ||SR

3) In quadrilateral PQRS

PQ=RS and PQ||RS (proved)

-: PQRS is a parallelogram

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