NCERT Solutions for Class 9th Mathematics

Chapter 8 – QUADRILATERALS

(Complete Downloadable Chapter Solution PDF file is at the bottom of the page)

 

EXERCISE 8.1

 

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

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

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

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

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

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.

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

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.

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

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

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.

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

 

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 = AC/2

   (ii) PQ = SR

   (iii) PQRS is a parallelogram.

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

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.

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.

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.

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

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 = AB/2