Question

What is the number of distinct triangles with integral valued sides and perimeter 14?

Answer 1

Hint: The perimeter of a triangle is the sum of all the sides and the sum of two sides of a triangle is always greater than the third side, using this concepts we will be able to the required result.

Complete step-by-step answer:
Given, the perimeter of the triangle is 14.
As, we know that the sum of two sides of a triangle is always greater than the third side.
Therefore, the sum of two sides will be more than the mean of the perimeter as the mean is the is the benchmark for the third number it cannot be greater than 7 $\left({\displaystyle \frac{14}{2}}=7\right)$
Now, we have to take the possibility of three numbers and also combination of them, which are as follows:
$\Rightarrow$ Let the sides be 4, 6, 4, then its perimeter is 14.
$\Rightarrow$ Let the sides be 5, 5, 4, then its perimeter is 14.
$\Rightarrow$ Let the sides be 3, 6, 5, then its perimeter is 14.
$\Rightarrow$ Let the sides be 6, 6, 2, then its perimeter is 14.
In all the above cases it is sure that the sum of two sides is always more than the third side, which satisfies the condition of the triangle.
Therefore, the number of possible distinct triangles will be 4 with integral valued sides and perimeter 14.

Note: Perimeter of a triangle is the outline length, i.e., the sum of all the sides of the triangle. There is one thing that needs to be known: the sum of two sides of a triangle is always greater than the third side.