Question

The lengths of the sides of a triangle is \[{\text{5 }}m\], \[1.2\] decameter and \[130{\text{ }}dm\]. Then its area is _______.
A. \[24{\text{ }}{m^2}\]
B. \[30{\text{ }}{m^2}\]
C. \[48{\text{ }}{m^2}\]
D. \[40{\text{ }}{m^2}\]

Answer 1

Hint: First of all, convert the lengths of sides of the triangle in meters. Now, use the formula, area of triangle \[ = \dfrac{1}{2} \times b \times h\] where \[b\] is the base of triangle and \[h\] is the height of the triangle in meters.

Complete step-by-step solution
Here, we have to find the area of the triangle having lengths of sides equal to 5m, \[1.2\] decameter and \[130{\text{ }}dm\].

We know that 1 decameter is \[10{\text{ }}m\] and \[{\text{1 }}dm = 0.1{\text{ }}m\]. Therefore, \[1.2\] decameter \[ = 1.2 \times 10 = 12{\text{ }}m\] and \[130{\text{ }}dm = 130 \times 0.1 = 13{\text{ }}m\].

So, the lengths of sides of the triangle in meters are \[{\text{5 }}m\], \[{\text{12 }}m\] and \[{\text{13 }}m\]. Let AB is \[{\text{5 }}m\], BC is \[{\text{12 }}m\] and CA is \[{\text{13 }}m\]. Then we can say that this triangle is a right-angled triangle because\[A{B^2} + B{C^2} = A{C^2}\].

\[
  A{B^2} + B{C^2} = A{C^2} \\
  {\left( 5 \right)^2} + {\left( {12} \right)^2} = {\left( {13} \right)^2} \\
  25 + 144 = 169 \\
  169 = 169 \\
\]

We know that the area of the triangle \[ = \dfrac{1}{2} \times b \times h\]…..(i) where \[b\] is the base of the triangle and \[h\] is the height of the triangle.
As we know that in a right angled triangle, the side with maximum length is hypotenuse. Therefore, CA is the hypotenuse of the triangle. We can take AB to be the height of the triangle and BC to be the base of the triangle.



Height of the triangle \[h\] is \[{\text{5 }}m\].
Base of the triangle \[b\] is \[{\text{12 }}m\].

So, by substituting the value of \[h = {\text{5 }}m\]and \[b = {\text{12 }}m\] in equation (i), we get, area of triangle \[ = \dfrac{1}{2} \times b \times h\]
\[
   = \dfrac{1}{2} \times 12 \times 5 \\
   = 6 \times 5 \\
   = 30{\text{ }}{m^2} \\
\]

So, we get the area of the triangle \[ = 30{m^2}\].
Therefore, the correct option is B.

Note: In this question, first of all, note that the units of length are different for all the sides. While calculating the area of a triangle all the length should be in the same unit. Also, identify the type of triangle before applying any formula because different types of triangles have different formulas to find area.