NCERT Solutions Class 9 Mathematics Chapter 8 Quadrilaterals Download In Pdf

Chapter 8 Quadrilaterals Download in Pdf

EXERCISE 8.1

**Question 1.** The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the
quadrilateral.

**Question 2**. If the diagonals of a parallelogram are equal, then show that it is a rectangle.

**Question 3.** Show that if the diagonals of a quadrilateral bisect each other at right angles, then it
is a rhombus.

**
Question 4.** Show that the diagonals of a square are equal and bisect each other at right angles.

**Question 5.** Show that if the diagonals of a quadrilateral are equal and bisect each other at right
angles, then it is a square.

**Question 6.** Diagonal AC of a parallelogram ABCD bisects
∠ A (see Fig. 8.19). Show that

(i) it bisects ∠ C also

(ii) ABCD is a rhombus.

**Question 7.** ABCD is a rhombus. Show that diagonal AC
bisects ∠ A as well as ∠ C and diagonal BD
bisects ∠ B as well as ∠ D.

**
Question 8.** ABCD is a rectangle in which diagonal AC bisects ∠ A as well as ∠ C. Show that:

(i) ABCD is a square

(ii) diagonal BD bisects ∠ B as well as ∠ D.

**Question 9.** In parallelogram ABCD, two points P and Q are
taken on diagonal BD such that DP = BQ
(see Fig. 8.20). Show that:

(i) Δ APD Δ CQB

(ii) AP = CQ

(iii) Δ AQB Δ CPD

(iv) AQ = CP

(v) APCQ is a parallelogram

**
Question 10.** ABCD is a parallelogram and AP and CQ are
perpendiculars from vertices A and C on diagonal
BD (see Fig. 8.21). Show that
:

(i) Δ APB Δ CQD

(ii) AP = CQ

**
Question 11.** In Δ ABC and Δ DEF, AB = DE, AB || DE, BC = EF
and BC || EF. Vertices A, B and C are joined to
vertices D, E and F respectively (see Fig. 8.22).
Show that
:

(i) quadrilateral ABED is a parallelogram

(ii) quadrilateral BEFC is a parallelogram

(iii) AD || CF and AD = CF

(iv) quadrilateral ACFD is a parallelogram

(v) AC = DF

(vi) Δ ABC Δ DEF.

**Question 12.** ABCD is a trapezium in which AB || CD and
AD = BC (see Fig. 8.23). Show that:

(i) ∠ A = ∠ B

(ii) ∠ C = ∠ D

(iii) Δ ABC Δ BAD

(iv) diagonal AC = diagonal BD
[Hint : Extend AB and draw a line through C
parallel to DA intersecting AB produced at E.]

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 Fig 8.29). AC is a diagonal. Show that :

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

(ii) PQ = SR

(iii) PQRS is a parallelogram.

**Question 2.** ABCD is a rhombus and P, Q, R and S are ©wthe mid-points of the sides AB, BC, CD
and DA respectively. Show that the quadrilateral PQRS is a rectangle.

**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.

**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 Fig. 8.30). Show that
F is the mid-point of BC.

**Question 5.** In a parallelogram ABCD, E and F are the
mid-points of sides AB and CD respectively
(see Fig. 8.31). Show that the line segments AF
and EC trisect the diagonal BD.

**Question 6.** Show that the line segments joining the mid-points of the opposite sides of a
quadrilateral bisect each other.

**Question 7.** ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB
and parallel to BC intersects AC at D. Show that:

(i) D is the mid-point of AC

(ii) MD ⊥ AC

(iii) CM = MA =
1
2
AB

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- Chapter 3 Coordinate Geometry
- Chapter 4 Linear Equations In Two Variables
- Chapter 7 Triangles
- Chapter 8 Quadrilaterals
- Chapter 9 Areas of Parallelograms and Triangles
- Chapter 10 Circles
- Chapter 11 Constructions
- Chapter 12 Heron’s Formula
- Chapter 13 Surface Areas and Volumes
- Chapter 14 Statistics
- Chapter 15 Probability

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