What will happen to the area of a rectangle if its length is doubled and breadth is same?

Question 6 Mensuration Exercise 20.4

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Answer:

(i) Length and breadth are trebled

Consider l as the initial length and b as the initial breadth

So the original area = l × b

If the length and breadth are trebled it becomes three times more than the original value

New length = 3l

New breadth = 3b

New area of the rectangle = 3l × 3b = 9lb

Hence, the area of the rectangle becomes 9 times more than its original area.

(ii) Length is doubled and breadth is same

Consider l as the initial length and b as the initial breadth

So the original area = l × b

If the length is doubled and breadth is same we get

New length = 2l

New breadth = b

New area of the rectangle = 2l × b = 2lb

Hence, the area of the rectangle becomes 2 times more than the original area.

(iii) Length is doubled and breadth is halved

Consider l as the initial length and b as the initial breadth

So the original area = l × b

If the length is doubled and breadth is halved we get

New length = 2l

New breadth = b/2

New area of the rectangle = 2l × b/2 = lb

Hence, the area of the rectangle does not change.

Video transcript

"hello guys welcome back to the room book i am rivan kumar working as a tutor that you do so today we are going to solve one interesting and very exciting topic so let's see what is that here in this figure that is in this figure so a b is parallel to c d and cd is parallel to ef and pq is parallel to rs right so this pq is parallel to rs okay so now observe properly rqd so r qd they have given it as 25 degrees and c q p so that is c q p they have given so that is of 60 degrees here so now q r s is equals to how much that is this total qrs is equals to how much is our question so we need to find this whole angle here so how we can find that now one thing observed properly as this is 25 that is okay so r q d they are given that is 25 degrees so in the same way a and r and q so a r q will be also 25 degrees because as they are vertically opposite angles okay so as they are vertically opposite angles here so they are equal that is 25 degrees and 25 degrees so now we should find only this angle because we got this is us 25 degrees right so now one more thing so angle c q p is equals to angle uh s r b so angle s r b that is this so we got here 60 60 degrees because because as when the two uh these two parallel lines see this is one parallel and this is one parallel line when something traversal is divided right so it will be equal because they are they are alternate exterior angles right so they are alternate exterior angles as they are equal here so now they are like 60 degrees so now we got here 60 and then what this will be so now as this is a straight angle that is a rb so angle are be this total okay this total is 180 degrees and out of this 180 degrees this is 60 we got now so the angle so this is 180 degrees here so as it is a straight line now angle a and r is plus angle srb is equals to 180 degrees is equals to 180 degrees now we got this srb is equal to 60 and this is x plus 60 degrees is equal to 180 then x is equals to 180 minus 60 is equals to 120 so this is the thing here so i hope you understand what i've done here if you have any doubts please comment below don't forget to subscribe to this channel so at last we got this is 120 plus 25 145 is our final answer so thank you so much for watching this thank you"

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In Mathematics, mensuration deals with geometric figures and parameters like volume, area, shape, surface area, etc. Or in other words, when we work with the area, the volume of specific shapes, or different parameters of geometric figures then it is called Mensuration in Mathematics.

Rectangle

A rectangle is a closed two-dimensional figure composed of four sides and four vertices. All angles of the rectangle are 90°. A rectangle with all sides equal is equivalent to a square. A rectangle is composed of two pairs of parallel sides, length, and width respectively.

Properties of rectangle

• A rectangle has four sides with four angles.
• The angles of a rectangle are the right angles that are 90° each.
• The opposite sides of a rectangle are parallel and equal in length.
• The diagonals of the rectangle bisect each other and both the diagonals have the same length.
• The sum of all the interior angles of a rectangle is equal to 360°.

Area of Rectangle

A rectangle is composed of equal pairs which are parallel in nature and equal in length. The area of a rectangle is the space enclosed within its boundaries. Or in other words, the product of the length and width of the rectangle is known as the area of a rectangle.

Area of rectangle = Length x Width

Let us assume A to be the area of the rectangle and l and b to be the length and breadth of the rectangle respectively.

A = l x b 

Solution: 

Let us assume A to be the original area of the rectangle. 

Let us assume l and b to be the length and breadth of the original rectangle respectively. 

Now, 

A = l x b 

Now, 

Let us assume l’ and b’ to be the length and breadth of the new rectangle respectively.  

A’ = l’ x b’ 

Now, the length is doubled and breadth remains the same, therefore, 

l’ = 2l 

b’ = b 

We get, 

A’ = 2l x b 

A’ = 2 (l x b)

A’ = 2A

Hence, when the length is doubled and the breadth remaining the same then the area of the rectangle becomes twice 

Sample Questions

Question 1: How does the area of the rectangle change when the length is doubled and breadth halved?

Solution:

We know,

Area of rectangle = length x breadth

Therefore,

A = l x b

Now,

l’ = 2l

b’ = b/2

Now, computing the area,

A’ = l’ x b’

A’ = 2l x b/2

A’ = l x b

Therefore, area remains same.

Question 2: How does the area change if length and breadth become equal?

Solution:

We know,

Area of rectangle = length x breadth

Therefore,

Area of rectangle = length x length

= (length)2

In this situation, the rectangle becomes a square.

Question 3: Derive the general formula of the change in the area of a rectangle if length becomes m times and breadth becomes n times.

Solution:

We know,

Area of rectangle, A = length x breadth

Therefore, in the modified case,

l ‘ = m x l

b’ = n x b

Area of rectangle, A’ =  l’ x b’

Area of rectangle, A’ =  m x l x n x b

Area of rectangle, A’ = m x n x (l x b)

A’ = m x n x A

Therefore, area becomes (m xn) times.

Question 4: Using the above formula, define how the area will change if length becomes 1/8 times and breadth 2 times.

Solution:

Area of rectangle = length x breadth

A’ = 1/8 x 2 A

= 1/4 A

Therefore, the area becomes one-fourth times of the original area.

Question 5: If the area becomes triple, keeping the length the same, how does the breadth change?

Solution:

Area = length x breadth

Area’ = 3 x Area

Now,

Length’ x Breadth’ = 3 x Length x Breadth

Given, length is constant, length’ = length

Therefore,

Breadth’ = 3 x Breadth

Thus, breadth becomes three times.