ABCD is a recharge P, Q, R, S are mid-point of a side AB, BC, CD, DA respectively. Show that the quadrilateral P, Q, R, S is a rhombus
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Answer:
Given :-
- ABCD is a rectangle and P, Q, R, S are mid-point of a side AB, BC, CD, DA respectively.
To Prove :-
- PQRS is a rhombus.
Construction:-
- Join diagonal AC.
Proof :-
□ In triangle ABC,
P is the midpoint of AB.
Q is the midpoint of BC.
So, by using midpoint theorem,
⟹ PQ || AC & PQ = 1/2 AC..........(1)
□ In triangle ADC,
R is the midpoint of CD.
S is the midpoint of DA.
So, by using midpoint theorem,
⟹ RS || AC & RS = 1/2 AC..........(2)
From (1) and (2)
PQ || RS & PQ = RS
⟹ PQRS is a parallelogram.
(If in a quadrilateral, if one pair of opposite sides are equal and parallel, then quadrilateral is a parallelogram)
□ Now, ABCD is a rectangle.
∴ AD = BC (Opposite sides of rectangle)
⟹ 1/2 AD = 1/2 BC
⟹ SD = QC
□ Now, In triangle SDR & triangle QCR
SD = QC (Proved above)
/_SDR = /_QCR (Each 90°)
DR = RC (R is the midpoint of CD)
∴ ️SDR ≅ ️QCR (SAS)
⟹ SR = RQ (CPCT)...........(3)
Also, PQRS is a parallelogram. (Proved above)
⟹ PQ = RS & PS = QR.......(4)
From (3) and (4), we concluded
PQ = QR = RS = SR
∴ PQRS is a rhombus.
Additional Knowledge
Properties of Parallelogram
- If a quadrilateral has a pair of parallel opposite sides, then it’s a special polygon called Parallelogram. The properties of a parallelogram are as follows:
- The opposite sides are parallel and congruent
- The opposite angles are congruent
- The consecutive angles are supplementary
- If anyone of the angles is a right angle, then all the other angles will be the right angle
- The two diagonals bisect each other
- Each diagonal bisects the parallelogram into two congruent triangles
- Sum of square of all the sides of parallelogram is equal to the sum of square of its diagonals. It is also called parallelogram law
Properties of Rhombus
- All sides of the rhombus are equal.
- The opposite sides of a rhombus are parallel.
- Opposite angles of a rhombus are equal.
- In a rhombus, diagonals bisect each other at right angles.
- Diagonals bisect the angles of a rhombus.
- The sum of two adjacent angles is equal to 180 degrees.
- The two diagonals of a rhombus form four right-angled triangles which are congruent to each other.
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