9th Maths

Quadrilaterals

Exercise 8.1

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

Answer: As you know angle sum of a quadrilateral = 360°

So, 3x + 5x + 9x + 13x = 360°
Or, 30x = 360°
Or, x = 12°
Hence, 3x = 36°, 5x = 60°, 9x = 108° and 13x = 156°




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

Answer; In the following parallelogram both diagonals are equal:

quadrilaterals 2
quadrilaterals 3

As all are right angles so the parallelogram is a rectangle.


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

quadrilaterals 4

Answer; In the given quadrilateral ABCD diagonals AC and BD bisect each other at right angle. We have to prove that AB=BC=CD=AD

quadrilaterals 5

So, AB=AD
Similarly AB=BC=CD=AD can be proved which means that ABCD is a rhombus.

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

Answer: In the figure given above let us assume that

quadrilaterals 6

DO=AO (Sides opposite equal angles are equal)
Similarly AO=OB=OC can be proved
This gives the proof of diagonals of square being equal.


Key Points About Quadrilaterals

  • Sum of the angles of a quadrilateral is 360°.
  • A diagonal of a parallelogram divides it into two congruent triangles.
  • In a parallelogram,
    • opposite sides are equal
    • opposite angles are equal
    • diagonals bisect each other
  • A quadrilateral is a parallelogram, if
    • opposite sides are equal or
    • opposite angles are equal or
    • diagonals bisect each other or
    • a pair of opposite sides is equal and parallel
  • Diagonals of a rectangle bisect each other and are equal and vice-versa.
  • Diagonals of a rhombus bisect each other at right angles and vice-versa.
  • Diagonals of a square bisect each other at right angles and are equal, and vice-versa.
  • 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.
  • A line through the mid-point of a side of a triangle parallel to another side bisects the third side.
  • The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram.