Question

How do you find the width of a rectangle with a diagonal of 12 centimeters and a length of 10 centimeters?

Answer 1

Hint: In a rectangle, all the angles are right angles. Also, the diagonal and two adjacent sides of a rectangle form a right-angled triangle. Because of this, we can use the Pythagoras Theorem to calculate any one side, if two others are known. The Pythagoras theorem states that the square of the hypotenuse is equal to the sum of the squares of the two other sides.

Complete step by step answer:
We are given the length of the diagonal and width of the rectangle and have to find the width of the rectangle. We know that it is a property of a rectangle that the diagonal and two adjacent sides of a rectangle form a right-angled triangle. As it is a right-angled triangle, Pythagoras theorem can be used for this triangle which states that the square of the hypotenuse (in this case, the diagonal) is equal to the sum of the squares of the two other sides (in this case, the length and the width).

Hence, using this property in the given rectangle. We get,
\[{{d}^{2}}={{l}^{2}}+{{w}^{2}}\]
\[d,l,w\]are the diagonal, length and width of the rectangle respectively.
Substituting the values, in the above equation, we get
\[\begin{align}
  & \Rightarrow {{\left( 12 \right)}^{2}}={{\left( 10 \right)}^{2}}+{{w}^{2}} \\
 & \Rightarrow 144=100+{{w}^{2}} \\
\end{align}\]
Subtracting 100 from both sides of above equation, we get
\[\begin{align}
  & \Rightarrow 144-100=100+{{w}^{2}}-100 \\
 & \Rightarrow {{w}^{2}}=44 \\
\end{align}\]
Taking square root of both sides, we get
\[\begin{align}
  & \Rightarrow w=\sqrt{44} \\
 & \Rightarrow w=\sqrt{2\times 2\times 11} \\
 & \Rightarrow w=2\sqrt{11} \\
\end{align}\]
Hence, width of the rectangle is \[2\sqrt{11}cm\].

Note:
 The properties of different types of quadrilaterals like square, rectangle, rhombus, parallelogram, etc. should be remembered. It will be useful to solve these types of problems. Calculation mistakes should be avoided.