Say true or false.
The opposite sides of a rectangle are equal in length.
(A) True
(B) False
Last updated date: 27th Mar 2023
•
Total views: 308.7k
•
Views today: 5.85k
Answer
308.7k+ views
Hint: Divide the rectangle into two parts diagonally. Then use Pythagoras theorem on the two triangles which are obtained after dividing diagonally. Then use the result to prove the lengths of the rectangle.
Complete step by step answer:
Complete step by step answer:
First let us consider a rectangle ABCD with AD as its diagonal, as shown below:
Now the diagonal divides the rectangle into two equal parts as shown in the figure above.
Now as we know the sides of the rectangle are perpendicular to each other. So the two triangles obtained after dividing the rectangle diagonally will be right angled triangles.
Now consider the right angled triangle ABD. By Pythagoras theorem we have the sum of square of two sides is equal to the square of the hypotenuse, that is,
$A{{D}^{2}}=A{{B}^{2}}+B{{D}^{2}}...........(i)$
Similarly, consider the right angled triangle ACD. By Pythagoras theorem we have the sum of square of two sides is equal to the square of the hypotenuse, that is,
$A{{D}^{2}}=A{{C}^{2}}+C{{D}^{2}}...........(ii)$
Now as we can observe in equation (i) and (ii), the left hand side is equal. So equating these two equations, we get
$ A{{B}^{2}}+B{{D}^{2}}=A{{C}^{2}}+C{{D}^{2}} $
$ \Rightarrow A{{B}^{2}}=A{{C}^{2}}+C{{D}^{2}}-B{{D}^{2}}........(iii) $
Now let us divide the rectangle ABCD using the other diagonal, i.e., BC, as shown below:
Now consider the right angled triangle ABC. By Pythagoras theorem we have the sum of square of two sides is equal to the square of the hypotenuse, that is,
$B{{C}^{2}}=A{{B}^{2}}+A{{C}^{2}}...........(iv)$
Similarly, consider the right angled triangle BCD. By Pythagoras theorem we have,
$B{{C}^{2}}=B{{D}^{2}}+C{{D}^{2}}...........(v)$
Now as we can observe in equation (iv) and (v), the left hand side is equal. So equating these two equations, we get
$ A{{B}^{2}}+A{{C}^{2}}=B{{D}^{2}}+C{{D}^{2}} $
$ \Rightarrow A{{B}^{2}}=B{{D}^{2}}+C{{D}^{2}}-A{{C}^{2}}........(vi) $
Now equating equation (iii) and (vi), we get
$A{{C}^{2}}+C{{D}^{2}}-B{{D}^{2}}=B{{D}^{2}}+C{{D}^{2}}-A{{C}^{2}}$
Cancelling like terms, we get
$ A{{C}^{2}}-B{{D}^{2}}=B{{D}^{2}}-A{{C}^{2}} $
$ \Rightarrow A{{C}^{2}}+A{{C}^{2}}=B{{D}^{2}}+B{{D}^{2}} $
$ \Rightarrow 2A{{C}^{2}}=2B{{D}^{2}} $
$ \Rightarrow A{{C}^{2}}=B{{D}^{2}} $
Taking square root on both sides, we get
AC = BD……..(vii)
Substituting this value in equation (iii), we get
$ A{{B}^{2}}=A{{C}^{2}}+C{{D}^{2}}-A{{C}^{2}} $
$ \Rightarrow A{{B}^{2}}=C{{D}^{2}} $
Taking square root on both sides, we get
$AB = CD$
Hence the opposite sides of a rectangle are equal in length. Therefore, the given statement is TRUE.
Therefore the correct answer is option (A).
Note: Another approach is considering diagonals of rectangle are equal. So, AD = BC.
Now consider the right angled triangle ABC and ABD and apply the Pythagoras theorem. Here these two triangles are similar. So, they will give exact answer which shows the opposite sides are equal in length.
Now the diagonal divides the rectangle into two equal parts as shown in the figure above.
Now as we know the sides of the rectangle are perpendicular to each other. So the two triangles obtained after dividing the rectangle diagonally will be right angled triangles.
Now consider the right angled triangle ABD. By Pythagoras theorem we have the sum of square of two sides is equal to the square of the hypotenuse, that is,
$A{{D}^{2}}=A{{B}^{2}}+B{{D}^{2}}...........(i)$
Similarly, consider the right angled triangle ACD. By Pythagoras theorem we have the sum of square of two sides is equal to the square of the hypotenuse, that is,
$A{{D}^{2}}=A{{C}^{2}}+C{{D}^{2}}...........(ii)$
Now as we can observe in equation (i) and (ii), the left hand side is equal. So equating these two equations, we get
$ A{{B}^{2}}+B{{D}^{2}}=A{{C}^{2}}+C{{D}^{2}} $
$ \Rightarrow A{{B}^{2}}=A{{C}^{2}}+C{{D}^{2}}-B{{D}^{2}}........(iii) $
Now let us divide the rectangle ABCD using the other diagonal, i.e., BC, as shown below:
Now consider the right angled triangle ABC. By Pythagoras theorem we have the sum of square of two sides is equal to the square of the hypotenuse, that is,
$B{{C}^{2}}=A{{B}^{2}}+A{{C}^{2}}...........(iv)$
Similarly, consider the right angled triangle BCD. By Pythagoras theorem we have,
$B{{C}^{2}}=B{{D}^{2}}+C{{D}^{2}}...........(v)$
Now as we can observe in equation (iv) and (v), the left hand side is equal. So equating these two equations, we get
$ A{{B}^{2}}+A{{C}^{2}}=B{{D}^{2}}+C{{D}^{2}} $
$ \Rightarrow A{{B}^{2}}=B{{D}^{2}}+C{{D}^{2}}-A{{C}^{2}}........(vi) $
Now equating equation (iii) and (vi), we get
$A{{C}^{2}}+C{{D}^{2}}-B{{D}^{2}}=B{{D}^{2}}+C{{D}^{2}}-A{{C}^{2}}$
Cancelling like terms, we get
$ A{{C}^{2}}-B{{D}^{2}}=B{{D}^{2}}-A{{C}^{2}} $
$ \Rightarrow A{{C}^{2}}+A{{C}^{2}}=B{{D}^{2}}+B{{D}^{2}} $
$ \Rightarrow 2A{{C}^{2}}=2B{{D}^{2}} $
$ \Rightarrow A{{C}^{2}}=B{{D}^{2}} $
Taking square root on both sides, we get
AC = BD……..(vii)
Substituting this value in equation (iii), we get
$ A{{B}^{2}}=A{{C}^{2}}+C{{D}^{2}}-A{{C}^{2}} $
$ \Rightarrow A{{B}^{2}}=C{{D}^{2}} $
Taking square root on both sides, we get
$AB = CD$
Hence the opposite sides of a rectangle are equal in length. Therefore, the given statement is TRUE.
Therefore the correct answer is option (A).
Note: Another approach is considering diagonals of rectangle are equal. So, AD = BC.
Now consider the right angled triangle ABC and ABD and apply the Pythagoras theorem. Here these two triangles are similar. So, they will give exact answer which shows the opposite sides are equal in length.
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

A Short Paragraph on our Country India
