Question

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 ​​

Answer 1

Answer:

$\huge\pink{\boxed{\blue{\boxed{ \purple{ \boxed{{\pink{Answer}}}}}}}} \\ \large\pink{\boxed{\blue{\boxed{ \purple{ \boxed{{\pink{Your~answer↓}}}}}}}}$

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.

$\boxed{ \large{ \mathfrak{Hence, proved.}}}$

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.