Question

Two sides of a triangle are 12cm and 7cm. Find the range for the length of its third side lies between 5cm and 19cm.

Answer 1

Hint: Using the property of inequality of sides in a triangle which states that the sum of two sides are always greater than the third side of the triangle. This property helps us in solving our question. We can easily find the range of the third side of the triangle.

Complete step-by-step answer:

First side of the triangle is 12cm, as given in the question.
Second side of the triangle is 7cm, as given in the question.
Now, we have to find the range of the third side using two given sides of the triangle as mentioned above.
As, we know that the inequality property of the triangle “In a triangle sum of two sides is always greater than the third side”.
Let, the third side of the triangle be the x centimeter in units.
Firstly, we have to calculate the sum of two sides i.e. first side and second side of the triangle.
Let, first side of the triangle be ‘a’.
Let, second side of the triangle be ‘b’.
So, sum = a + b = 12 + 7 = 19 cm.
Hence, we obtained the sum of two sides of the triangle to be 19cm.
Then, $a+b>x$
$x<19\ldots (1)$
It means that the value of x should be less than 19.
Also, we have another relation,
$7+x>12$
As, the sum of two sides of the triangle must be greater than 12 i.e. first side of triangle.
$x>5\ldots (1)$
It means that the value of x should be greater than 5.
So, from equation (1) and equation (2) we get,
$x\in (5,19)$
Hence, the value of x must lie between 5 and 19.

Note: The key step in this problem is the concept of inequality of sides of the triangle. After that step, the problem is reduced to mathematical calculations only. The mathematical calculations must be performed accurately with the rules for solving inequalities in mind otherwise we may get an error.