Question

Calculate the area of the triangle whose sides are 18cm, 24cm and 30cm in length. Also find the length of the altitude corresponding to the smallest side.
A) $216\,c{m^2},24cm$
B) $220\,c{m^2},25cm$
C) $225\,c{m^2},25cm$
D) $168\,c{m^2},25cm$

Answer 1

Hint: Since we are given three sides of a triangle, so we will use the Heron’s Formula first to find the area of the triangle and then using the area of the triangle so obtained we will find the length of the altitude corresponding to the smallest side.

Complete step by step answer:

Now, according to the question, let the triangle be $\Delta ABC$, such that the three sides are a = 18cm, b = 24cm and c = 30cm.
Then by applying the Heron’s Formula we first need to find the semi-perimeter which is
$s = \dfrac{{a + b + c}}{2}$
So, substituting the values of a, b and c we will get:
$
  s = \dfrac{{18 + 24 + 30}}{2} \\
   = \dfrac{{72}}{2} \\
   = 36cm \\
 $
Next, using Heron’s Formula we will get:
Area of $\Delta ABC$=
$
  \sqrt {s(s - a)(s - b)(s - c)} \\
   = \sqrt {36(36 - 18)(36 - 24)(36 - 30)} \\
   = \sqrt {36 \times 18 \times 12 \times 6} \\
   = \sqrt {6 \times 6 \times 3 \times 6 \times 2 \times 6 \times 6} \\
   = 6 \times 6 \times 6 \\
   = 216\,c{m^2} \\
 $
Now, to calculate the length of the perpendicular corresponding to the smallest side,
Smallest side of the triangle is 18 cm.
Since area of the triangle will be the same,
Also, we know that Area of $\Delta ABC$=$\dfrac{1}{2} \times base \times height$,
So Area of $\Delta ABC$= $216\,c{m^2}$, and base is 18cm so substituting these values we will get the height or the length of the perpendicular from the shortest side of 18 cm we get:
$
  216 = \dfrac{1}{2} \times 18 \times Height \\
   \Rightarrow 216 = 9 \times Height \\
   \Rightarrow Height = \dfrac{{216}}{9} \\
   \Rightarrow Height = 24cm \\
 $
Therefore, the area of the given triangle is $216\,c{m^2}$and the length of the altitude corresponding to the smallest side of 18cm, is 24cm.

Hence, the correct answer is option A.

Note: The formula for finding areas of different triangles is different. We use Heron’s formula to find the areas of triangles,in which none of the sides are equal. Now perimeter of a triangle is the sum of all three sides, while semi-perimeter is the half of this perimeter, denoted by: $s = \dfrac{{a + b + c}}{2}$ where a,b and c are the three respective sides of the triangle. The usual formula used for calculating the area of the triangle is $\dfrac{1}{2} \times base \times height$.