Question

The area of a triangular region is 12,692 m\[^2\]. The perpendicular from one vertex to the opposite side is 76 m. Find the length of this side.

Answer 1

Hint:
First, we will find the dimension of the triangle. Then use the formula of area of the triangle is \[A = \dfrac{1}{2}bh\], where \[b\] is the base of the triangle and \[h\] is the height of the triangle. Apply this formula of area of a triangle, and then use the given conditions to find the required value.

Complete step by step solution:
It is given that the area \[A\] of triangular region \[{\text{ABC}}\] is 12,692 m\[^2\] and the height of this triangle is 76 m.
Let us assume that the breadth of the triangular region is \[b\] m.
We will now draw the triangle ABC using the given condition.

We know that the area of the triangle is calculated as \[A = \dfrac{1}{2}bh\], where \[b\] is the base of the triangle and \[h\] is the height of the triangle.
We will now substitute the value of the area \[A\] of the triangle \[{\text{ABC}}\] and the height of the triangle \[h\] in the above formula.
\[
  12,692 = \dfrac{1}{2} \times b \times 76 \\
   \Rightarrow 12,692 = 38b \\
 \]
Dividing the above equation by 38 into each of the sides to find the base of the triangular region, we get
\[
   \Rightarrow \dfrac{{12,692}}{{38}} = \dfrac{{38b}}{{38}} \\
   \Rightarrow b = 334{\text{ cm}} \\
 \]

Thus, the base of the triangular region is 334 cm.

Note:
In solving these types of questions, first draw the pictorial representation of the given problem for better understanding. Some students use the formula of area of the triangle is \[Area = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \], where \[s\] is the semi perimeter of the triangle and \[a\], \[b\] and \[c\] are the sides of the triangle instead of the one we used, which is wrong. Since we are given in the question that there is a perpendicular from one vertex implies that the formula, \[A = \dfrac{1}{2}bh\], where \[b\] is the base of the triangle and \[h\] is the height of the triangle will be used for finding the area of triangle.