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8. Quadrilaterals

Class 9 Mathematics Chapter 8. Quadrilaterals

Chapter 8. Quadrilaterals

Important Note :

1. Sum of the angles of a quadrilateral is 360°.
2. A diagonal of a parallelogram divides it into two congruent triangles.
3. In a parallelogram , opposite sides are equal .

4. In a parallelogram, opposite angles are equal
5. In a parallelogram, diagonals bisect each other .
6.  A quadrilateral is a parallelogram, if opposite sides are equal .

7.  A quadrilateral is a parallelogram, if opposite angles are equal .  

8.  A quadrilateral is a parallelogram, if diagonals bisect each other .
9. A quadrilateral is a parallelogram, if a pair of opposite sides is equal and parallel .
10. Diagonals of a rectangle bisect each other and are equal and vice-versa.
11. Diagonals of a rhombus bisect each other at right angles and vice-versa.
12. Diagonals of a square bisect each other at right angles and are equal, and vice-versa.
13. The line-segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of it.
14. A line through the mid-point of a side of a triangle parallel to another side bisects the third side.
15. The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram. 

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EXERCISE 8.1

1. The angle of quadrilateral are in the ratio  . Find all the angles of the quadrilateral .

Solution: let, and  are the angle of quadrilateral respectively .

We know that , the sum of the angles of a quadrilateral is 360° .

      

     

    

    

       

        

       

And 

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

Solution: let ABCD is a parallelogram and AC = BD . Then we show that ABCD is a rectangle .

Proof : Since, ABCD is a parallelogram , then

AB = CD  and AD = BC

    

In and, we have

AB=AB  [ Common side ]

  AD=BC  [Given]

  BD=AC [Given]

 [SSS]

   [CPCT]

Again,  and  is a transversal .

 

 

So, ABCD is a parallelogram in which one angle is 90° .

Therefore, ABCD is a rectangle .

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

Solution:  let ABCD is a quadrilateral such that ,

and  . AC and BD are the diagonals . Then we show that quadrilateral ABCD is rhombus .

Proof: In figure :

    

In  and,we have

    

 

    

    [SAS]

  [CPCT]

Similarly, , and

So,  

Therefore, the quadrilateral ABCD is rhombus .

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

Solution: let ABCD is a square and the diagonals AC = BD . Then we show that

 

Proof : Since, ABCD is a square .

    OA = OC and  OB=OD

AB = BC = CD = AD 

In  and ,we have

 

 

  

     [SSS]

    [CPCT]

Now, 

 

  

  

  

So, 

 and  .

    Proved .

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

Solution: let ABCD is a quadrilateral and AC = BD and  .

Then we show that ABCD is a square .

Proof: Since , the diagonals AC and BD bisect , then

OA = OC and OB = OD

In  and,we have

         [given]

   

    [common side]

     [SAS]

   [CPCT]

Similarly , , and

So, AB = BC = CD = AD

In  and ,we have

    [given]

  

   [common side]

     [SAS]

       [CPCT]

Again, 

     

  

   

       

  So,  

Similarly , 

and

  

Therefore, ABCD is a square .

6. Diagonal AC of a parallelogram ABCD bisects  (see Fig. 8.19) . Show that (i) it bisects  also,  (ii) ABCD is a rhombus .

         

Solution: Given, the diagonal AC of a parallelogram ABCD bisects   . Then we show that :

 (i) it bisects  also,

(ii) ABCD is a rhombus .

Proof:   Since, ABCD is a parallelogram .

So, AB = CD  and AD = BC

And AC is the bisector of  .

 

(i)  In and  , we have

     [common side]

       [given]

      [given]

     [SAS]

   

So, is the bisector of  .

Therefore. AC bisect  .  Proved.

(ii)  In and  , we have

    [common side]

     [given]

       [given]

     [SAS]

      [CPCT]

 

Therefore , ABCD is a rhombus .

7. ABCD is a rhombus . Show that diagonal AC bisects  as well as  and diagonal BD bisects  and  .

Solution:  Given,  ABCD is a rhombus . Then we show that diagonal AC bisects  as well as  and diagonal BD bisects  and  .

Proof : Since, ABCD is a rhombus .

  So, AB = BC = CD = AD

   

In and  , we have

     [common side]

     [given]

    [given]

   [SSS]

 [CPCT]

So, AC bisect

and  

  So,  AC bisect  . 

Therefore, the diagonal AC bisects  as well as  .

Second part :

In and , we have

   [common side]

    [given]

    [given]

    [SSS]

  [CPCT]

So, BD bisect

and  

  So,  BD bisect  . 

Therefore, the diagonal BD bisects  as well as  .

8. ABCD is a rectangle in which diagonal AC bisects  as well as  . Show that : (i) ABCD is a square  (ii) diagonal BD bisects  as well as  .

Solution:  Given, ABCD is a rectangle in which diagonal AC bisects  as well as  .Then we show that : (i) ABCD is a square  (ii) diagonal BD bisects  as well as  .

Proof: Since, ABCD is rectangle .

So, AB = CD and AD = BC

 

(i) We have, AC bisects  .

 

and AC bisects

In and  , we have

    [common side]

      [given]

     [given]

    [SSS]

  [CPCT]

So, AB = BC = CD = AD

and   

Therefore, ABCD is a square .  Proved .

(ii)   In and  , we have

    [common side]

     [given]

    [given]

   [SSS]

  [CPCT]

So, BD bisect

and  

  So,  BD bisect  . 

Therefore, the diagonal BD bisects  as well as  .

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

     

(i)     (ii)    (iii)     (iv)   (v) is a parallelogram .

Solution:  Given, In parallelogram ABCD , two points P and Q are taken on diagonal BD such that  . Then we show that : (i)     (ii)     (iii)     (iv)    (v)  is a parallelogram .

Proof: Since, ABCD is a parallelogram .

So,   and

  and

 (i) In   and  , we have

             [Given]

    [Alternative interior angle]

    [Given]

   [SAS]

                                Proved.

(ii)  In  and  , we have

        [Given]

   [Alternative interior angle]

        [Given]

    [SAS]

     [CPCT]

Proved.

(iii)  In  and , we have

        [Given]

  [Alternative interior angle]

       [Given]

     [SAS]

                                         Proved.

(iv) In  and , we have

        [Given]

   [Alternative interior angle]

      [Given]

     [SAS]

     Proved.

(v)   Proof: Since,   [SAS]

       

And     [SAS]

    

 We know that a quadrilateral is a parallelogram, then a pair of opposite sides is equal and parallel .

So, and  

Again,  and  

Thus, APCQ is a parallelogram .

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

 

 (i)     (ii)   

Solution:   Given, ABCD is a parallelogram and AP and CQ are perpendicular from vertices A and C on diagonal BD . Then we show that :  (i)     (ii)   

Proof: (i)  In  and  , we have

                [ ABCD is a parallelogram]

          

          [Alternative interior angle ]

          [ASA]

(ii)  In  and  , we have

           [ ABCD is a parallelogram]

    

    [Alternative interior angle ]

    [ASA]

        [CPCT]

11. In and  ,  and  . 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)  and  

(iv) quadrilateral ACFD is a parallelogram 

(v) 

(vi) 

 

Solution:  Given, and ,  and . Vertices A , B and C are joined to vertices D , E and F respectively .

Then we  show that :

(i) quadrilateral ABED is a parallelogram

(ii) quadrilateral BEFC is a parallelogram

(iii)  and  

(iv) quadrilateral ACFD is a parallelogram 

(v) 

(vi) .

Proof: In ABED quadrilateral .

We know that, a quadrilateral is a parallelogram , when a pair of opposite angle is equal and parallel .

So,   and

Therefore, the quadrilateral ABED is a parallelogram .

(ii)  In BEFC quadrilateral .

We know that, a quadrilateral is a parallelogram , when a pair of opposite angle is equal and parallel .

So, and

Therefore, the quadrilateral BEFC is a parallelogram .

(iii) Since, the quadrilateral ABED is a parallelogram .

So, and  

Again, the quadrilateral BEFC is a parallelogram .

So, and

From  and  we get

   and     Proved.

(iv) In ACFD quadrilateral  .

We know that, a quadrilateral is a parallelogram , when a pair of opposite angle is equal and parallel .

So,  and

Therefore, the quadrilateral ACFD is a parallelogram .

Proved.

(v)  Since, the quadrilateral ACFD is a parallelogram .

So,  and

Therefore, AC = DF

                                Proved.

(vi)  In  and  , we have

 AB=DE [given]

  BC=EF [given]

  AC=DF  [given]

  [SSS]     Proved.

12. ABCD is a trapezium in which  and   (see Fig. 8.23) . Show that :

   

(i)  

(ii)   

(iii) 

(iv) diagonal  diagonal  .

[ Extend AB and draw a line through C parallel to DA intersecting AB produced at E] 

Solution:  Given, ABCD is a trapezium in which  and  .Then we show that

(i) 

(ii)   

(iii)

(iv) diagonal  diagonal  .

Construction:  Extend AB and we draw a line through C parallel to DA intersecting AB produced at E .

Proof:  Since,  

And AD = BC , then AD = BC = CE .

In  , we have

   

So, BCE is an isosceles triangle.

  

Again,  and  is transversal .

 

  

 

and  [Linear pair of angles]

   

From  and  , we get

 

 

       Proved.

(ii) Since,  and  is transversal .

  

Again, and is transversal .

 

From (1) and (2) , we get

  

     [  ]

     

Proved.

(iii) In and ,we have

     [common side]

   [given]

  

  [SAS]

(iv) In and ,we have

 [common side]

  [given]

  

    [SAS]

 [CPCT]

Therefore, Diagonal AC = Diagonal BD .  Proved.

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)  and     (ii) PQ = SR    (iii) PQRS is a parallelogram.

Solution:  Given, ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA . AC is a diagonal.

Then we show that :  (i)   and   (ii) PQ = SR       (iii) PQRS is a parallelogram.

Proof: (i)  Since, S and R are the mid-point of the sides AD and CD of a  respectively .

  and       Proved.

(ii) Since, S and R are the mid-point of the sides AD and CD of a  respectively .

 and

Again, Since, P and Q are the mid-point of the sides AB and BC of a  respectively .

 and 

From(i) and (ii) , we get     Proved.

(iii) Since, S and R are the mid-point of the sides AD and CD of a  respectively .

    and

Again, Since, P and Q are the mid-point of the sides AB and BC of a  respectively .

 and

From(i) and (ii) , we get

    and 

We know that , a quadrilateral is a parallelogram then a pair of opposite sides is equal and parallel .

Thus, PQRS is a parallelogram .     Proved.

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.

Solution:  Given, ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Then we show that the quadrilateral PQRS is a rectangle.

Proof: Since, P and Q are the mid-point of the sides AB and BC of a  respectively .

   and

Again, S and R are the mid-point of the sides AD and CD of a  respectively .

     and

From(i) and (ii) , we get  

Similarly,

So, PQRS is a parallelogram .

Since, ABCD is rhombus .

AB = BC = CD = AD

And AP = BP = BQ = CQ = CR = DR = AS = DS

In  ,We have

     

So, APS is an isosceles triangle .

 

In  and  , we have

BP = DR  [Given ]

  [Given]

BQ = DS  [Given]

     [SAS]

    [CPCT]

We have,

  [Linear pair of angles]

 and

 From (i) and (ii) , we get

 

   [  and ]

Since,  and  is a transversal .

 

So, PQRS is a parallelogram in which one angle is 90° .

Therefore, PQRS is a rectangle . Proved

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: Given, ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Then we show that the quadrilateral PQRS is a rhombus.

Proof: Since, P and Q are the mid-point of the sides AB and BC of a  respectively .

      and

Again, S and R are the mid-point of the sides AD and CD of a  respectively .

  and

From(i) and (ii) , we get   

So, PQRS is a parallelogram .

Since, ABCD is rectangle .

So, AP = BP [P is the mid-point of AB]

And AS = BQ  [S and Q is the mid-point of AD and BC]

In and , we have

         [Given]

      

         [Given]

     [SAS]

          [CPCT]

 

Hence , PQRS is a rhombus . Proved.

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.

Solution: Given, ABCD is a trapezium in which , 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 . Then we show that F is the mid-point of BC.

Proof:  Since, ABCD is a trapezium and  .

Let BD and EF intersecting at M .

By the mid-point theorem, we get

  and E is the mid-point of AD .

So,   and M is the mid-point of BD .

 Also

In   and    

So, F is the mid-point of BC .    Proved.

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.

Solution: Given,  a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively . Then we show that the line segments AF and EC trisect the diagonal BD.

Proof: Since, E and F are the mid-points of sides AB and CD respectively .

So, and

ABCD is a parallelogram .

So, and

 and  AE∥CE 

Hence, AECF is a parallelogram .

So,  

By the mid-point  theorem ,

In  ,we have

Since, E is a mid-point of AB and  ,then Q is the mid-point of BP .

 

In  ,we have

Since, F is a mid-point of CD and  ,then P is the mid-point of DQ .

From (i) and (ii) , we get

     

Hence, the line segments AF and EC trisect the diagonal BD.  Proved.

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

Solution:  Given, PQRS is a quadrilateral and A, B, C and D are mid-points of the sides PQ, QR, RS and PS respectively .Then we show that the diagonals AC and BD bisect each other .

Proof:  Since, A and B are the mid-point of the sides PQ and QR of a  respectively .

    and

Again, Since, C and D are the mid-point of the sides RS and PS of a  respectively .

   and

From(i) and (ii) , we get

  and 

Similarly,   and

 and

From (iii) and (iv), we get

 and

Thus, ABCD is a parallelogram .

We know that the diagonals of a parallelogram bisect each other .

Hence, the diagonals AC and BD 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)  (iii) .

Solution: Given, 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.

Then we show that (i) D is the mid-point of AC      (ii)       (iii)

Proof: In figure:

(i) In , we have

Since, M is the mid-point of hypotenuse AB and .

By the mid-point theorem, we get

D is the mid-point of the side AC .  Proved

(ii)  Given,

Since,

So,  [ Corresponding angle]

And  also

Thus,    Proved.

(iii) In   and  , we have

        [D is a mid-point of AC]

      [Common side]

 

  [RHS]

 [CPCT]

So, 

  

   

 

  Proved .