Question

What will be the area of the triangle formed by the line $$9x + 4y = 36$$ with both of the axes?
1). 9 sq units
2). 18 sq units
3). 27 sq units
4). 36 sq units

Answer 1

Hint: Here in this question, we have to find the area of the triangle formed by the given line. For this, first we need to plot the graph of given line then mark the triangle inside the graph with both of the axis next find the area of triangle by using formula $$Area = \dfrac{1}{2} \times base \times height$$, then substitute the value of base and height by the graph on further simplification using the basic multiplication operation we get the required area of triangle.

Complete step-by-step solution:
The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle. Basically, it is equal to half of the base times height, i.e., $$Area = \dfrac{1}{2} \times base \times height$$ or $$A = \dfrac{1}{2} \times b \times h$$. Hence, to find the area of a tri-sided polygon, we have to know the base (b) and height (h) of it.
Consider the equation of given line:
 $$9x + 4y = 36$$ ----(1)
Put $$x = 0$$, then
$$ \Rightarrow \,\,\,9\left( 0 \right) + 4y = 36$$
$$ \Rightarrow \,\,\,4y = 36$$
Divide 4 on both side, then we get
$$ \Rightarrow \,\,\,y = 9$$
$$\therefore \,\,\left( {x,y} \right) = \left( {0,9} \right)$$
Similarly
Put $$y = 0$$, then
$$ \Rightarrow \,\,\,9x + 4\left( 0 \right) = 36$$
$$ \Rightarrow \,\,\,9x = 36$$
Divide 9 on both side, then we get
$$ \Rightarrow \,\,\,x = 4$$
$$\therefore \,\,\left( {x,y} \right) = \left( {4,0} \right)$$
The graph of the equation of line $$9x + 4y = 36$$ is given by

In the above graph the shaded region represents a triangle $$\Delta \,AOB$$.
The area of $$\Delta \,AOB$$ is:
  $$ \Rightarrow \,\,Area = \dfrac{1}{2} \times OA \times OB$$
$$ \Rightarrow \,\,Area = \dfrac{1}{2} \times 4 \times 9$$
$$ \Rightarrow \,\,Area = \dfrac{1}{2} \times 36$$
$$\therefore \,\,Area\,of\,\Delta \,AOB = 18\,\,sq\,units.$$
Therefore, option (2) is the correct answer.

Note: While determining the area we use the formula. The formula is $$A = \dfrac{1}{2} \times b \times h$$. The unit for the perimeter will be the same as the unit of the length of a side or triangle. Whereas the unit for the area will be the square of the unit of the length of a triangle. We should not forget to write the unit.