Question

In the figure given below, find the area of \[\vartriangle ABC\] (in square units):-

Answer 1

Hint: We will find the height and base of the triangle using the coordinate axis and then just use the formula $Area = \dfrac{1}{2} \times base \times height$ by putting in the values and getting the required answers.

Complete step-by-step answer:
We can clearly see the figure as shown in the graph given above.
The base of the triangle lies on the x- axis which can be termed as BC.
The height of the triangle need to be constructed by drawing an altitude AD perpendicular to BC, then AD will be the height.
The new figure is as shown below:

Here, now we have both altitude and base. We now need to find their lengths.
We can see from the figure that coordinates of A, B, C and D are (1, 3), (-1, 0), (4, 0) and (1, 0) respectively.
Let us first discuss the distance formula:-
Let the two points be $({x_1},{y_1})$ and $({x_2},{y_2})$. So, the distance d between between the given two points are given by: $d = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} $. This distance is known as Euclidean distance. This is an application of Pythagorean Theorem.
Hence, the length of BC is $BC = \sqrt {{{(4 + 1)}^2} + {{(0 - 0)}^2}} = \sqrt {{{(5)}^2}} = 5units$ ……….(1)
And the length of AD is $AD = \sqrt {{{(1 - 1)}^2} + {{(0 - 3)}^2}} = \sqrt {{{( - 3)}^2}} = 3units$ …………(2)
Now, we will put in (1) and (2) in the following formula because BC is the base and AD is the height.
$Area = \dfrac{1}{2} \times base \times height$
$ \Rightarrow Area = \dfrac{1}{2} \times 5 \times 3 = \dfrac{{15}}{2}unit{s^2} = 7.5unit{s^2}$.
Hence, the answer is 7.5 square units.

Note: The students must notice that we drew D straight without any calculations because A’s coordinates were given. So, we already know the x coordinate of D.
Alternate way:- You may find the length of each side using distance formula and then use the heron’s formula to find the area of the triangle.
It is given by $Area = \sqrt {s(s - a)(s - b)(s - c)} $, where s is the semi perimeter of the triangle, a, b and c are the side lengths of all the sides of the triangle.