Question

$ABC$ is a triangle with the length of two of its sides equal to $15.6$ centimeters and $10.6$ centimeters and $15.6$ centimeters is the longest side length of the triangle. The perimeter of the triangle is _____.

• A. less than $30.2$ centimeters
• B. greater than $31.2$ centimeters and less than $52.4$ centimeters
• C. less than $5$ centimeters
• D. greater than $4$ centimeters and less than $50$ centimeters

Answer 1

The correct option is B greater than $31.2$ centimeters and less than $52.4$ centimeters

We know that the sum of the length of any two sides of a triangle must be greater than the length of the third side.

Let the length of the third side be $a$ cm.
According to the condition, $10.6+a>15.6$
$\Rightarrow a>5...\left(1\right)$


Also,
$a<10.6+15.6\Rightarrow a<\mathrm{26.2.........}\left(2\right)$

The length of the third side cannot be greater than $26.2cm$ since $a<26.2cm.$
Therefore, the third side should be greater than $5cm$ and less than $26.2cm.$


Perimeter of the triangle
$=15.6+10.6+a=26.2+a$

But we know $a>5.$
Therefore, $26.2+a>26.2+5$
$\Rightarrow$ Perimeter of the triangle $>31.2cm$ …………(3)

We also know $a<\mathrm{26.2.}$
Therefore, $26.2+a<26.2+26.2$
$\Rightarrow$ Perimeter of the triangle $<52.4cm$ …………(4)

From (3) and (4) we have,
$31.2cm<\text{perimeter}<52.4cm$