class IX Quadrilateral

 

QUADRILATERALS

1.Two parallel lines l and m are intersected by a transversal p (see Fig.). Show that the quadrilateral formed by the bisectors of interior angles is a rectangle.

 

 

2.Show that the bisectors of angles of a parallelogram form a rectangle.

3.Show that if the diagonals of a quadrilateral are equal and bisect each other at right.

 

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

(i)               

∆ APD ≅ ∆ CQB

(ii)              AP = CQ

(iii)            ∆ AQB ≅∆ CPD

(iv)            AQ = CP

(v)              APCQ is a parallelogram

5. In ∆ ABC and ∆ 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.). 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)            ∆ ABC ≅ ∆ DEF.

 

6. ABCD is a trapezium in which AB || CD and AD = BC (see Fig.). Show that

 (i) ∠ A = ∠ B

 (ii) ∠ C = ∠ D

(iii) ∆ ABC ≅ ∆ BAD

 (iv) diagonal AC = diagonal BD

7. l, m and n are three parallel lines intersected by transversals p and

 q such that l, m and n cut off equal intercepts AB and BC on p (see

 Fig.). Show that l, m and n cut off equal intercepts DE and EF on q

 also

 

 

 

 

 

8. ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig). AC is a diagonal. Show that:

(i)               

SR || AC and SR = 1/ 2 AC

(ii)              PQ = SR

(iii)           PQRS is a parallelogram.

 

 

 

9. 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.

 

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

                                                             

 

 

 

 11. 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