Quadrilaterals: class 9 math: 9 Math

NCERT Exercise 8.2 solution: 9th math

Quadrilaterals Class 9 math NCERT Exercise 8.2 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 figure). AC is a diagonal.

Show that

(i) SR||AC and SR = `1/2`AC

(ii) PQ = SR

(iii) PQRS is a parallelogram

Solution

Given, ABCD is a quadrilateral.

And P, Q, R and S are mid points of sides AB, BC, CD and DA respectively.

And, AC is a diagonal

Thus, to prove that

(i) SR||AC and SR =`1/2`AC

In triangle ACD,

Points S and R are the mid points of sides AD and DC

Thus according to the Mid-point Theorem which says that the line segment joining the mid-points of any two sides of a triangle is parallel and half to the third side.

Therefore, SR||AC - - - - - (i)

And, SR = `1/2`AC - - - - (ii)

Therefore, SR||AC and SR = `1/2`AC Proved

To prove (ii) PQ = SR

In triangle ABC,

Points P and Q are mid points of side AB and BC respectively.

Thus according to the Mid-point Theorem which says that the line segment joining the mid-points of any two sides of a triangle is parallel and half to the third side.

PQ || AC - - - - - (iii)

And, PQ = `1/2` AC - - - - - (iv)

Now, from equation (ii) and equation (iv), we get

PQ = SR Proved

To prove (iii) PQRS is a parallelogram

From equation (i) and equation (iii), we get

PQ||SR

And, from equation (ii) and equation (iv), we get

PQ = SR

Now, since one pair of opposite sides of the quadrilateral PQRS are parallel and equal, consequently other pair of opposite sides will also be equal.

Thus, PQRS is a parallelogram. Proved

Quadrilaterals Class 9 math NCERT Exercise 8.2 Question (2) ABCD is a rhombus and P, Q, R and S are mid points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.

Solution

Let ABCD is the given rhombus.

And let P, Q, R and S are the mid points of side AB, BC, CD and DA

Thus, to prove that PQRS is a rectangle.

This means here it is to be proved that, PQRS is a parallelogram and one of its internal angle is a right angle.

Now, In triangle DBC,

Q and R are mid-points of BC and CD

Therefore, according to Mid-Point Theorem

RQ||DB - - - - - - (i)

And, RQ = `1/2`DB - - -- - (ii)

Now, In triangle ABD,

P and S are mid-points of AB and AD

Therefore, according to Mid-Point Theorem

PS||DB - - - - - (iii)

And, PS = `1/2`DB - - - - - (iv)

Now, from equation (i) and equation (iii), we get

PS||RQ

And, PS = RQ

Now, since one pair of opposite sides in quadrilateral PQRS are parallel and equal, thus, PQRS is a parallelogram.

Now, Since, ABCD is a Rhombus,

Thus, it's diagonals intersect each other at right angle.

This means ∠MON = 90^o

Now, in quadrilateral ONQM,

Since, RQ||DB

Therefore, MQ||ON

Thus, ONQM is a parallelogram.

And, ∠MON = ∠NQM

[Because opposite angles of a parallelogram are equal]

Now, since one of the angles of parallelogram PQRS is a right angle.

i.e. ∠NQM = ∠PQM = 90^o

Therefore, PQRS is a rectangle. Proved

Quadrilaterals Class 9 math NCERT Exercise 8.2 Question (3) ABCD is a rectangle and P, Q, R, and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a Rhombus.

Solution

Let ABCD is the given rectangle.

In this rectangle, P, Q, R and S are mid-points of side AB, BC, CD and DA

Thus, to prove that PQRS is a Rhombus

Proof

In triangle ABC,

P is the mid-point of side AB

And, Q is the mid-point of side BC

Therefore, according to Mid-Point Theorem which says that The line segment joining the mid-points of two sides of a triangle is parallel and half of the third side.

Therefore,

PQ||AC - - - - - (i)

And, PQ = `1/2`AC - - - - (ii)

And, In triangle ACD,

Point S is the mid-point of side AD

And, R is the mid-point of DC

Therefore, according to Mid-Point Theorem

SR||AC - - - - - (iii)

And, SR = `1/2` AC - - - - - (iv)

Now, from equation (i) and equation (ii), we get

PQ||SR

And, from equation (ii) and equation (iv), we get

PQ = SR - - - - (v)

Now, since one pair of opposite sides of quadrilateral PQRS is parallel and equal, thus PQRS is a parallelogram.

This means another pair of opposite sides will also be parallel and equal.

That means QR||PS

And QR = PS - - - - (vi)

Now, in triangle ABD,

P and S are mid-points of sides AB and AD

Therefore, according to Mid-Point Theorem

PS = `1/2`BD

⇒ PS = `1/2`AC - - - - (vii)

[Because AD and BC are diagonals of rectangle ABCD and diagonals of rectangle are equal]

Now, from equation (ii) and equation (vi) we get

PS = PQ - - - - (viii)

Thus, from equation (v), (vi) and (viii), we get

PQ = QR = SR = PS

Since all sides of parallelogram PQRS is equal, thus, PQRS is a Rhombus Proved

Quadrilaterals Class 9 math NCERT Exercise 8.2 Question (4) ABCD is a trapezium in which AB||DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see figure). Show that F is the mid-point of BC.

Solution

Given, ABCD is a trapezium.

In which AB||DC

And BD is its diagonal.

E is the mid-point of side AD

EF||AB

Then to prove point F is the mid-point of BC

Proof

Let line EF intersects BC at point O

In triangle ABE,

E is the mid-point of side AD [As given in question]

And EF||AB [As given in question]

Therefore, according to mid-point theorem which says that if a line is drawn through the mid-point of any side of a triangle and parallel to another side, it bisects the third side.

Therefore, O is the mid-point of side BD

Now, Since AB||DC and AB||EF

Therefore, EF||DC

[Because according to one of the theorem of parallel lines we know that, Lines which are parallel to the same line are parallel to each other]

Now, in triangle BCD,

O is the mid-point of line BD [As already proved above]

And, EF||DC

Therefore, according to mid-point theorem which says that if a line is drawn through the mid-point of any side of a triangle and parallel to another side, it bisects the third side.

OF bisects the line BC

This means F is the mid-point of BC Proved

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