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Home / Geometry / Solution of the activity| Quadrilaterals| Class eight(8)

Solution of the activity| Quadrilaterals| Class eight(8)

Quadrilaterals: A quadrilateral is a closed figure bounded by four line segments.

Type of Quadrilaterals:

Parallelogram: A parallelogram is a quadrilateral with opposite sides parallel. The region bounded by a parallelogram is also known as parallelogram.

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Rectangle: A rectangle is a parallelogram with a right angle .The region bounded by a rectangle is a rectangular region.

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Rhombus: A rhombus is a parallelogram with equal adjacent sides ,i.e. ,the opposite sides of rhombus are parallel and the lengths of four sides are equal.  Region bounded by a rhombus is also called rhombus.

 

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Square: A square is a rectangle with equal adjacent sides, .e., a square is a parallelogram with all sides equal and all right angles. The area bounded by square is also called a square.

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Trapezium: A trapezium is a quadrilateral with a pair of parallel sides. The region bounded by trapezium is also called trapezium.

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(a)Area of trapezium

Construct DA∥CE at C.

∴AECD is a parallelogram. From the figure                    Screenshot_18

area of trapezium=area of parallelogram

AECD + area of triangle CEB

=a × h +b – a × h

=b + a h

∴ Area of trapezium=average of the sum of two parallel sides ×height.

(b)Area of Rhombus: The diagonals of a rhombus bisect each other at right angles. If we know the lengths of two diagonals, we can find the area of rhombus.

Let the diagonals AC and BD of a rhombus

ABCD intersect each other at O.Screenshot_19

Denote the lengths of two diagonals by

a and b respectively.

Area of rhombus= area of triangle ∆DAC + ∆BAC.

=1/2 a×1/2 b+1/2 a×1/2b

=1/2a×b

∴ Area of rhombus =half of the product of two diagonals.

Ex: Prove that every angle of a rectangle is right angle.

General enunciation:  Prove that every angle of a rectangle is right angle.

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Particular enunciation: Let ABCD is a rectangle in which ∠A=1 right angle.

We have to prove that every angle of a rectangle is right angle i.e. ∠A=∠B=∠C=∠D=1 Right angle

Proof:  We know that, Rectangle is a parallelogram and opposite angles of a parallelogram equal to each other

∴∠A=∠C and ∠B=∠D   —————— (i)

Again, since rectangle is parallelogram

∴AB∥CD, AD is their transversal.

∴∠A+∠D=2 right angle   [sum of two interior angles on the same side

of a transversal is 2 right angle]

Or, ∠D=2 right angle – ∠A

=2 right angles – 1 right angle

 

∴∠D=1 right angle

From (i) we can write,

∠A=∠C= 1 right angle

And ∠B=∠D=1right angle

∴∠A=∠B=C=∠D=1 right angle (Proved)

Ex:1.Prove that the diagonals of a square are equal and bisect each other.

General enunciation: To prove that the diagonals of a square are equal and bisect each other.

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Particular enunciation: Let ABCD is a square whose the diagonals AC and BD insect each other at O. We have to prove that (i)AC=BD (ii)AO=CO,BO=DO.

Proof: We know, square is a parallelogram and the diagonals of a parallelogram bisect each other

∴AO=CO, BO=DO

In ∆ABD and ∆ACD,

AB=DC                          [sides of equal to each other]

And AD=AD                 [common side]

Included ∠BAD =included ∠ADC    [each of the angles is right angle]

∴∆ABD≌∆ACD                      [SAS theorem]

Therefore AC=BD (proved)

Ex: a worker has made a rectangular concrete slab. In how many different ways can he be sure that the slab is really rectangular?

Solution: In the following way the worker can be sure that the slab is really rectangular.

  1. If the opposite sides of the slab are equal and parallel.
  2. If every angle of the slab is right angle.
  3. If its two diagonals are equal and bisect each other.

Ex: Find the area of trapezium by an alternate method.

Solution: Let ABCD is a trapezium in which the parallel sides are AB = a unit and DC =b unit respectively. We join A,C and draw a perpendicular CE on the side AB from the point C.

Let CE= h unit.Screenshot_22

Therefore Area of the trapezium ABCD

 

=(Area of ∆region ABC)+(Area of ∆region ADC)

=(1/2 ×AB×CE)square unit +(1/2 ×DC×CE)square unit    [the height of ∆ADC is CE]

=1/2 ×CE × (AB+DC) square unit

=1/2 h (a +b  ) square unit.

Check Also

If two triangles have the three sides of the one equal to the three sides of the other, each to each, then they are equal in all respects.

 If two triangles have the three sides of the one equal to the three sides ...

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