Find perimeter of a square if its diagonal is \[10\sqrt{2}\] 10 cm \[40\sqrt{2}\]cm 20 cm 40 cm
Chapter 2 – Pythagoras Theorem- Text Book Solution
Problem Set 2 | Q 1.5 | Page 43
Some question and their alternative answer are given. Select the correct alternative.
Find perimeter of a square if its diagonal is \[10\sqrt{2}\]
10 cm
\[40\sqrt{2}\]cm
20 cm
40 cm
It is given that ABCD is a square.
∴ AB = BC = CD = DA = x (say)
According to Pythagoras theorem, in ∆ABD
\[{\text{AB}}^2 + {\text{AD}}^2 = {\text{BD}}^2 \]
\[ \Rightarrow x^2 + x^2 = \left( 10\sqrt{2} \right)^2 \]
\[ \Rightarrow 2 x^2 = 200\]
\[ \Rightarrow x^2 = 100\]
\[ \Rightarrow x = \sqrt{100}\]
\[ \Rightarrow x = 10 \text{cm}\]
Hence, the side of the square is 10 cm.
Now,
Perimeter of a square = \[4 \times \left( side \right)\]
=\[4 \times x\]
=\[4 \times 10\]
=\[40\]
Hence, the correct option is 40 cm.
Explanation:-
The problem involves finding the perimeter of a square when the length of its diagonal is known. It is given that ABCD is a square and all its sides are equal, i.e., AB = BC = CD = DA = x. Using Pythagoras theorem in ∆ABD, we get:
AB^2 + AD^2 = BD^2 x^2 + x^2 = (10√2)^2 (since BD = 10√2, as given in the problem) 2x^2 = 200 x^2 = 100 x = √100 x = 10 cm
Hence, the side of the square is 10 cm. To find the perimeter of the square, we use the formula:
Perimeter of a square = 4 x (side)
Substituting the value of x, we get:
Perimeter of the square = 4 x 10 = 40
Therefore, the perimeter of the square is 40 cm.
Chapter 2 – Pythagoras Theorem- Text Book Solution
Problem Set 2 | Q 1.5 | Page 43