Perimeter and Area

Triangles as Parts of Rectangles

Let two rectangles which are shown here:

rectangle square

Here A is a rectangle, and diagonal is cutting this rectangle in two equal halves. Both the triangles are congruent. Hence their area will be equal.

Therefore,

Area of rectangle = Area of one triangle + Area of another triangle
⇒Area of rectangle = 2 × Area of one triangle (As both the triangles are equal)

`2 xx 1/2 xx` length `xx` breadth = length `xx` breadth

Similarly, For figure B, which is a square and diagonals cut that in four equal triangles. Means all triangles are congruent.

Area of square `= 2 xx` Area of triangle
`= 2 xx 1/2 xx` Side2
= Side2


Generalising for other Congruent Parts of Rectangles

Let us consider the rectangle given in the figure. In this a Line EF is dividing the rectangle in two equal part. Both parts are congruent.

rectangle

Hence, area of one part = area of other part.
Hence, area of each congruent part = Area of rectangle ÷ 2


In text solution based on Area(NCERT)

Question 1: Each of the following rectangles of length 6 cm and breadth 4 cm is composed of congruent polygons. Find the area of each polygon.

rectangle

Solution: Figure A has 6 congruent polygons.

Hence, area of each polygon = Area of rectangle ÷ 6
Area of rectangle = length X breadth
`= 6 xx 4 = 24` sq cm

Hence, area of each polygon of A `= 24 ÷ 6 = 4` cm2

Figure B has 4 congruent polygons.

Hence, area of each polygon = Area of rectangle ÷ 4
`= 24 ÷ 4 = 6` sq cm

Figure C has 2 congruent polygons.

Hence, are of each polygon = Area of rectangle ÷ 2
`= 24 ÷ 2 = 12` sq cm

Figure D has 2 congruent polygons.

Hence, are of each polygon = Area of rectangle ÷ 2
`= 24 ÷ 2 = 12` sq cm

Figure E has 8 congruent polygons.

Hence, area of each polygon = Area of rectangle ÷ 8
`= 24 ÷ 8 = 3` sq cm

AREA OF A PARALLELOGRAM

A polygon is said to be a parallelogram when their opposite sides are parallel.

parallelogram

Here; let ABCD is a parallelogram. In this AB is parallel to CD and AC is parallel to BD. One side BD of this parallelogram is extended and a perpendicular CE is drawn on it.

Here CE is called the Height of the parallelogram.

Hence;
Area of parallelogram ABCD = base x height
⇒ Area of a parallelogram = base x height.


Question 2: Find the area of following parallelograms:

parallelogram

(i) Given, Base = 8cm, Height = 3.5 cm

Therefore, Area = Base x Height
`= 8cm \xx 3.5cm = 28.0 cm^2`

(ii) Given, Base = 8cm, Height = 2.5 cm

Therefore, Area = Base x Height
`= 8cm \xx 2.5cm = 20.0 cm^2`

(iii) In a parallelogram ABCD, AB = 7.2cm and the perpendicular from C on AB is 4.5cm.

parallelogram

Area of parallelogram ABCD = Base x Height

`= 7.2 cm \xx 4.5 cm = 32.40 cm^2`



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