Answer

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**Hint:**In this problem, we have to find the perimeter of the given triangle.

The given value of two sides of the triangle. From that, we have to find each side value of the triangle. Using the equation, at first, we will find the length of each side.

We know that the perimeter of the triangle is the total length of the sides of the triangle.

Then adding the length of the sides, we get the perimeter of the triangle.

**Complete step by step answer:**

We have to find the perimeter of \[\vartriangle ABC\].

At first, we will find the length of each side of the given triangle.

Here,

\[AB + BC = 10\] cm ………………... (1)

\[BC + CA = 12\] cm …………….…... (2)

\[CA + AB = 16\] cm …………………. (3)

Adding (1), (2) and (3) we get,

\[\Rightarrow 2(AB + BC + CA) = 10 + 12 + 16\] cm

Simplifying we get,

\[\Rightarrow 2(AB + BC + CA) = 38\] cm

Dividing we get,

\[\Rightarrow (AB + BC + CA) = 19\] cm …………….. (4)

Subtracting (4) and (1) we get,

\[\Rightarrow CA = 19 - 10 = 9\] cm

Subtracting (4) and (2) we get,

\[\Rightarrow AB = 19 - 12 = 7\] cm

Subtracting (4) and (3) we get,

\[\Rightarrow BC = 19 - 16 = 3\] cm

We know that the perimeter of any triangle is the total length of the sides of the triangle.

So, the perimeter of \[\vartriangle ABC\] is \[AB + BC + CA\] cm

Substitute the values we get,

$\Rightarrow $ The perimeter is \[= 9 + 3 + 7 = 19\] cm.

**$\therefore $ The perimeter of the given triangle is \[19\] cm. Hence the correct option is option (A).**

**Note:**

In this, we are finding the perimeter of the triangle. We know that the sum of the lengths of the sides is the perimeter of any polygon. Therefore, in the case of a triangle, the perimeter will be the sum of all the three sides. If a triangle has three sides $a$, $b$, and $c$, then, perimeter $= a + b +c$. But the given in this problem is the value of the addition of the two sides. So we have to calculate each side value of the triangle. In that calculation, students may go mistakenly. So, they have to focus on those calculations.

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