8. Quadrilaterals
NCERT
8. Quadrilaterals
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Important Note :
Chapter 8. Quadrilaterals
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.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
EXERCISE 8.1
1. The angle of quadrilateral are in the ratio 
. 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 angle , then it 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 .
7. ABCD is a rhombus . Show that diagonal AC bisects as well as and diagonal BD bisects 
and 
.
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 .
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 .
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)
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) 
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]
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.
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) 
(iii) 
.
Posted 3 years ago