Question

The area of the square , one of whose diagonals is y $3\hat i + 4\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} $ is
A.12 square units
B.12.5 square units
C.25 square units
D. 156.25 square units

Answer 1

Hint: We are given one of the diagonals of the square. Firstly we need to find the length of the diagonal by using modulus formula (i.e.) Modulus of a vector $a\hat i + b\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} = \sqrt {{a^2} + {b^2}} $ . With the length of the diagonal we can find the area using the formula $\frac{1}{2}{d^2}$ where d is the length of the diagonal.

Complete step-by-step answer:
Step 1:
We are given that one of the diagonals is $3\hat i + 4\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} $.
We known that the length of the diagonal can be obtained by finding the modulus of the vector $3\hat i + 4\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} $
Modulus of a vector $a\hat i + b\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} = \sqrt {{a^2} + {b^2}} $
So now ,
$\begin{gathered}
   \Rightarrow \left| {3\hat i + 4\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} } \right| = \sqrt {{3^2} + {4^2}} \\
  {\text{ }} = \sqrt {9 + 16} \\
  {\text{ = }}\sqrt {25} = 5{\text{ }}units \\
\end{gathered} $
Step 2:
Now we have that the length of the diagonal is 5 units
Now the area of the square is given by $\frac{1}{2}{d^2}$ sq. units where d is the length of the diagonal
Area =$\frac{1}{2}{d^2}$
 =$\frac{1}{2}*{5^2}$
 =$\frac{1}{2}*25$
 =12.5 sq units.
Therefore the area of the square is given by 12.5 sq units
The correct option is B


Note: Generally , the area of a quadrilateral is given by half of the product of their diagonals.
Since it is a square the diagonals are equal , hence the formula is $\frac{1}{2}{d^2}$.
If a side is given then the area of the square would be ${a^2}$
So students must be very attentive in reading the question as many tend to miss the half in the formula when a diagonal is given.