Question

The sides of a \[8 \times 8\] square are cut by certain points into pieces of length 1 and 7, 2 and 6, 3 and 5, and 4 and 4 respectively. The area of the quadrilateral determined by these four points is
(a) 28
(b) 36
(c) 48
(d) 8
  

Answer 1

Hint: Here, we need to find the area of the quadrilateral. First, we will find the area of the square. Then, we will find the area of the four triangles. Finally, we will use these areas to find the area of the quadrilateral formed by joining the points.

Formula Used: We will use the following formulas:
The area of a square is given by the formula \[{s^2}\], where \[s\] is the length of the side of the square.
The area of a triangle is given by the formula \[\dfrac{1}{2}bh\], where \[b\] is the base of the triangle, and \[h\] is the height of the triangle.

Complete step-by-step answer:
First, we will name the triangles and the quadrilateral formed.

We need to find the area of the quadrilateral E.
First, we will find the area of the square and the four triangles.
The area of a square is given by the formula \[{s^2}\], where \[s\] is the length of the side of the square.
Substituting \[s = 8\] in the formula, we get
\[ \Rightarrow \]Area of the square \[ = {8^2}\]
Applying the exponent on the base, we get
\[ \Rightarrow \]Area of the square \[ = 64\]
Now, we will find the area of the triangle A, B, C, and D.
The area of a triangle is given by the formula \[\dfrac{1}{2}bh\], where \[b\] is the base of the triangle, and \[h\] is the height of the triangle.
Substituting \[b = 1\] and \[h = 4\] in the formula, we get
\[ \Rightarrow \]Area of triangle A \[ = \dfrac{1}{2}\left( 1 \right)\left( 4 \right)\]
Simplifying the expression, we get
\[ \Rightarrow \]Area of triangle A \[ = 2\]
Substituting \[b = 7\] and \[h = 2\] in the formula, we get
\[ \Rightarrow \]Area of triangle B \[ = \dfrac{1}{2}\left( 7 \right)\left( 2 \right)\]
Simplifying the expression, we get
\[ \Rightarrow \]Area of triangle B \[ = 7\]
Substituting \[b = 3\] and \[h = 6\] in the formula, we get
\[ \Rightarrow \]Area of triangle C \[ = \dfrac{1}{2}\left( 3 \right)\left( 6 \right)\]
Simplifying the expression, we get
\[ \Rightarrow \]Area of triangle C \[ = 9\]
Substituting \[b = 5\] and \[h = 4\] in the formula, we get
\[ \Rightarrow \]Area of triangle D \[ = \dfrac{1}{2}\left( 5 \right)\left( 4 \right)\]
Simplifying the expression, we get
\[ \Rightarrow \]Area of triangle D \[ = 10\]
Now, we will find the region of the quadrilateral E.
From the figure, we can observe that the area of the square is the sum of the areas of the four triangles and the quadrilateral.
Therefore, we get
Area of square \[ = \] Area of triangle A \[ + \] Area of triangle B \[ + \] Area of triangle C \[ + \] Area of triangle D \[ + \] Area of quadrilateral E
Substituting the areas of the four triangles and the square, we get
\[ \Rightarrow 64 = 2 + 7 + 9 + 10 + \]Area of quadrilateral E
Adding the terms, we get
\[ \Rightarrow 64 = 28 + \]Area of quadrilateral E
Subtracting 28 from both sides of the equation, we get
\[ \Rightarrow \]Area of quadrilateral E \[ = 64 - 28 = 36\]
Therefore, the area of the quadrilateral formed by joining the points is 36.
Thus, the correct option is option (b).

Note: We used the heights 4, 2, 6, 4 for triangles A, B, C, and D, respectively, because the four triangles are right angled triangles. This is because the four interior angles of a square are right angles. Therefore, the triangles A, B, C, and D are right angled triangles. Right angled triangles are those triangles whose one angle is \[90^\circ \].