Question

Find the diagonal of a square whose side is 14 cm.

Answer 1

Hint: In this question, we will divide a square into two right angled triangles and then apply Pythagoras theorem to one of them to find the length of the diagonal.

Complete step-by-step answer:
Let us consider a square ABCD with each side of length 14 cm. Let us now draw a diagonal AC by joining point A and C of the square.

Now, in a square, all its four sides are equal in length. And here, all four sides of the square are of the length 14 cm.
Therefore, $$\text{AB=BC=CD=DA=14 cm}$$.
 And, all the angles of a square are ${90}^{\circ }$.
Therefore, $\mathrm{\angle }\text{ABC}={90}^{\circ }$.
Also, according to Pythagoras theorem, in a right-angled triangle PQR, right angled at Q,$\text{P}{\text{Q}}^{\text{2}}\text{+Q}{\text{R}}^{\text{2}}\text{=P}{\text{R}}^{\text{2}}$.
Now, considering a triangle ABC in a square ABCD.
It is right angled at B, so applying Pythagoras theorem here, we get,
$\text{A}{\text{B}}^{\text{2}}\text{+B}{\text{C}}^{\text{2}}\text{=A}{\text{C}}^{\text{2}}$
Putting the values of AB and BC here, we get,
$\begin{array}{rl}& \text{1}{\text{4}}^{\text{2}}\text{c}{\text{m}}^2\text{+1}{\text{4}}^{\text{2}}\text{c}{\text{m}}^2\text{=A}{\text{C}}^{\text{2}}\\ & \Rightarrow \text{A}{\text{C}}^{\text{2}}\text{=}\left(\text{1}{\text{4}}^{\text{2}}\text{+1}{\text{4}}^{\text{2}}\right)\text{c}{\text{m}}^{\text{2}}\\ & \Rightarrow \text{A}{\text{C}}^{\text{2}}\text{=2}\times \text{1}{\text{4}}^{\text{2}}\text{c}{\text{m}}^{\text{2}}\end{array}$
Taking square root on both sides of the equation here, we get,
$\begin{array}{rl}& \sqrt{\text{A}{\text{C}}^{\text{2}}}\text{=}\sqrt{\text{2 }\times \text{ 1}{\text{4}}^{\text{2}}\text{c}{\text{m}}^{\text{2}}}\\ & \Rightarrow \sqrt{\text{A}{\text{C}}^{\text{2}}}\text{=}\sqrt{\text{2}}\sqrt{\text{1}{\text{4}}^{\text{2}}\text{c}{\text{m}}^{\text{2}}}\end{array}$
Here, cancelling square root with whole square, we get,
$\text{AC=14}\sqrt{\text{2}}\text{cm}$.
Hence, the diagonal of a square whose side is 14 cm is $14\sqrt{2}$ cm.

Note: This question can also be done directly with the formula that, in all squares, the length of a diagonal is always $\sqrt{2}$ times the length of the side of the square.