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The area of triangle ABC is \[20c{m^2}\]. The coordinates of vertex A are (−5,0) and those of B are (3,0). The vertex C lies on the line \[x - y = 2\]. The coordinates of C can be
a) (5, 3)
b) (-3, -5)
c) (-5, -7)
d) (7, 5)

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Last updated date: 13th Jun 2024
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Answer
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Hint:
A triangle with vertices at $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$. If the triangle was a right-angled triangle, it would be pretty easy to compute the area of a triangle by finding one-half the product of the base and the height (area of triangle formula). However, when the triangle is not a right-angled triangle there are multiple different ways to do so. It turns out that the area of triangle formula can also be found using determinants.
Area of triangle \[ = \dfrac{1}{2} \times \left| {\begin{array}{*{20}{c}}{x_1}&{y_1}&1\\{x_2}&{y_2}&1\\{x_3}&{y_3}&1\end{array}} \right|\]

Complete step by step solution:
Let any point on the line \[x - y = 2\] be \[C = (h,h - 2)\]
Coordinate of vertex \[A = ( - 5,0)\]
Coordinate of vertex \[B = (3,0)\]
Area of ..
\[\begin{array}{l} \Rightarrow \dfrac{1}{2} \times \left| {\begin{array}{*{20}{c}}h&{h - 2}&1\\{ - 5}&0&1\\3&0&1\end{array}} \right| = 20\\ \Rightarrow 8(h - 2) = 40\\ \Rightarrow h - 2 = \pm 5\\ \Rightarrow h = 7or - 3\end{array}\]
Hence, the points are (7, 5) and (-3, -5).

So, Option B and D are correct.

Note:
In order to find the area of a triangle, one must multiply the base by the height. Afterward, one must divide it by 2. The division by 2 comes because one can divide a parallelogram into 2 triangles.
In order to find the area of any right triangle, multiplication of the lengths of the two sides must take place. These two sides are perpendicular to each other. Afterward, one must take half of that.
A scalene triangle is one which has all three sides of different lengths. The area of a scalene triangle that has a base b and height h is given by 1/2 bh. If one knows the lengths of all three sides, one can find the area by making use of the Heron’s Formula without the need to find the height.