Question

ABCD is a square. F is the mid-point of AB.BE is one third of BC. If the area of $\vartriangle FBE = 108c{m^2}$, find the length of $AC$.

Answer 1

Hint: Here use properties of square and Pythagoras theorem to solve the question.
A square is a regular quadrilateral, which has all the four sides of equal length and all the four angles are also equal. The angles of the square are at right angles or equal to 90-degrees. The diagonals of the square are equal and bisect each other at 90 degrees.
Pythagoras theorem: Pythagoras theorem is a fundamental relation in Euclidean geometry among three sides of a right angle triangle. According to Pythagoras theorem, the area of the square whose sides is the hypotenuse is equal to the sum of the areas of the two squares on the other two sides.
Line bisector: A bisector perpendicular to the line divides the line in two equal parts.

Complete step-by-step solution:
Step: 1 Let side of the square is $a$.
Since square has equal sides,
$
  AB = BC \\
   \Rightarrow BC = CD \\
   \Rightarrow CD = AB
 $
According to question,
$BF = \dfrac{a}{2}$ and $BE = \dfrac{a}{3}$
$ \Rightarrow $Area of the triangle $FBE = \dfrac{1}{2} \times FB \times BE$
Step: 2 Substitute the value of $\left( {FB = \dfrac{a}{2}} \right)$ and $\left( {BE = \dfrac{a}{3}} \right)$in the formula,
$
  108 = \dfrac{1}{2} \times \dfrac{a}{2} \times \dfrac{a}{3} \\
   \Rightarrow {a^2} = 1296 \\
   \Rightarrow a = 36
 $
Step: 3 consider the triangle $ABC$ and use Pythagoras theorem.
According to Pythagoras theorem,
$A{C^2} = A{B^2} + B{C^2}$
Substitute the $\left( {AB = a} \right)$ and $\left( {BC = a} \right)$ in the formula.
$
  A{C^2} = {a^2} + {a^2} \\
   \Rightarrow A{C^2} = 2{a^2} \\
   \Rightarrow AC = \sqrt 2 a \\
 $
Substitute the value of $\left( {a = 36} \right)$ in the equation.
$
  AC = \sqrt 2 \times 36 \\
   \Rightarrow AC = 1.41 \times 36 \\
   \Rightarrow AC = 50.904
 $
Therefore the length of AC=50.904.

Therefore the length of side AC=50.904.

Note: Students are likely to remember the properties of squares. They must use the formula step by step carefully. Find the area of the right triangle (FBE) by using the formula $\left( {FBE = \dfrac{1}{2} \times FB \times BE} \right)$ to calculate the sides of the square. Use the Pythagoras theorem $\left( {A{C^2} = A{B^2} + B{C^2}} \right)$ for the triangle (ABC) to find the length of hypotenuse of triangle ABC in square ABCD.