Question

A triangle has sides $8cm,17cm$and $15cm$ The area of the triangle is:
A) $50c{m^2}$
B) $68c{m^2}$
C) $40c{m^2}$
D) $60c{m^2}$

Answer 1

Hint:If the length of sides of the triangle are $a,b$ and $c$ then the area of the triangle can be evaluated by using the Heron’s formula.
Heron’s formula is:
$\sqrt {s(s - a)(s - b)(s - c)} $, here $s$ is the semi-perimeter of the triangle or $s = \dfrac{{a + b + c}}{2}$

Complete step-by-step answer:
We are given that a triangle has sides $8cm,17cm$ and $15cm$.
We have to find the area of the triangle.
Let the length of sides of the triangle are $a,b$ and $c$.
According to the question,
$a = 8,b = 15$ and $c = 17$
In order to find the area of the triangle we use Heron’s formula.
The Heron’s formula states that if the length of sides of the triangle are $a,b$ and $c$ then the area of the triangle will be
$\sqrt {s(s - a)(s - b)(s - c)} $, here $s$is the semi-perimeter of the triangle or $s = \dfrac{{a + b + c}}{2}$
Before applying the formula, we evaluate the value of $s$.
Substitute the value of $a,b$ and $c$ in $s = \dfrac{{a + b + c}}{2}$ to evaluate the value of $s$.
$
   \Rightarrow s = \dfrac{{8 + 15 + 17}}{2} \\
   \Rightarrow s = 20cm \\
 $
Use the Heron’s formula to evaluate the area of the triangle.
\[
   \Rightarrow \sqrt {s(s - a)(s - b)(s - c)} \\
   = \sqrt {20(20 - 8)(20 - 15)(20 - 17)} \\
   = \sqrt {20 \times 12 \times 5 \times 3} \\
   = \sqrt {3600} \\
   = 60c{m^2} \\
 \]
Therefore, the area of the triangle is $60c{m^2}$.
Hence, option (D) is correct.

Note:
We can solve this question by another method which is shown below:
Evaluate the square of each side and establish a relation between them.
$
  {17^2} = 289 \\
  {15^2} + {8^2} = 225 + 64 = 289 \\
 $
It means the square of the length of the longest side is equal to the sum of the squares of the remaining sides.
We know that, By Pythagoras theorem which states that the square of the hypotenuse is equal to the sum of the square of the base and altitude.
Therefore, according to the Pythagoras theorem, it is the right-angled triangle with measurements:
Hypotenuse is $17cm$, base is $15cm$ and altitude is $8cm$.
We know that the area of the triangle is half of the product of the base and the altitude.
Therefore, Area of the triangle will be:
$\dfrac{{15 \times 8}}{2} = 60c{m^2}$

Hence, option (D) is correct.