Scalene Perimeter

In a triangle, the sum of the length any 2 sides must always be larger than the length of the 3rd.

Hence, the length of $AC$ must be less than $5$ and more than $1$ .

Since $ABC$ is a scalene triangle, $AC$ cannot be of length $2$ or $3$ .

Since $ABC$ has integer side lengths, $AC$ must be of length $4$ .

Therefore the perimeter is $2 + 3 + 4 = 9$ .

The sum of the length of two sides of a triangle is always greater than the third side. Hence, third side of the triangle should be less than 3+2= 5. But we can't choose 3 and 2 as the above triangle is scalene. Therefore the remaining options are 1 and 4. We can't choose 1 as 1+2=3, which is equal to 3(the length of third side). Hence 1 is discarded and 4 is the third side. Hence, perimeter is 2+3+4= 9.

The measure of the third side is strictly between $|2-3| = 1$ and $2+3=5.$ Since the triangle is scalene and has integer side lengths, the only possible length of the third side is $4.$ Hence, the perimeter is $2+3+4 = 9.$

Here we see that the two sides of the triangle ABC are 2 and 3 therefore sum of the two side AB and BC=2+3=5 Now we know that sum of two sides of a triangle is greater than the third side therefore 5 is greater than the third side possibilities of the length of the third side=1 or 2 or 3 or 4 but the third side cannot be 2 or 3 because the triangle is scalene therefore the third side is either 1 or 4 but if the third side is 1 then 1+2=3 i.e. the sum of two sides of triangle ABC is equal to the third side this is not possible therefore the third side = 4 therefore AB=2 , BC=3 and CA=4 therefore perimeter of triangle ABC=2+3+4 =9 Ans: The perimeter of triangle ABC is 9

Firstly, by the triangle inequality, we see that the length of the third side (c) $1<c<5$

Thus $c=2, 3,$ or $4$ .

Since the triangle is scalene, c cannot equal to 2 or 3, else it would be isosceles.

Thus $c=4.$

Perimeter $=2+3+4=9$

Call the length of the unknown side x.

From the triangle inequality we have 2+3>x, x<5 We also have x+2>3, x>1

As x is an integer, we have x=2,3,4

But 2,3 would repeat side lengths, so x must be 4.

Perimeter=2+3+4=9

a,b,c be sides with usual notations a=3 , c=2 a+c>b which implies 5>b 2,3 cannot be chosen as triangle is scalene and 1 does not satisfy triangle inequality 2+1=3 so 4 is the only choice left

as triangle is non-degenerate it can't have length of third side 0,1 or 5 because in these cases the third side lies on other sides. and length cannot be more than 5 because no non-degenerate triangle possible such that "sum of its two sides are less than or equal to third side". the possibility for third side is only 2, 3 or 4. as triangle is scalene so 2 and 3 not possible answers.so third side is 4. now 2+3+4=9 is perimeter.

Through the Triangle Inequality, the difference between 2 sides must not be more than the third side.

CA>AC-AB --> CA>1 AB>CA-BC --> 2>CA-3 --> CA<5.

Since ABC is scalene, CA could not be 2 or 3. Thus, CA is 4. Its perimeter is then 9.

We know that a triangle with side a, b, and c has a "trick" which a plus b is more than c, a plus c more than b, and b plus c more than a with a, b, and c are different. Since a is 2, b is 3, so 2 plus 3 more than c, c plus 3 more than 2, and c plus 2more than 3. So, c is 4.

Let $CD = x$ . By the triangle inequality we get the next $3$ inequalities:

$3 + 2 > x,\ 2 + x> 3,\ 3 + x > 2$

The inequalities imply that $5 > x > 1$ . As $x$ is integer, it can only take the values $2$ , $3$ or $4$ . But $2$ and $3$ are not possible, since the triangle is scalene.

Therefore $x = 4$ , and the perimeter is $2 + 3 + 4 = 9$ .

First of all, everyone should know that in any non-degenerate triangle, every side is smaller than the sum of the other two. This limits the length of $AC$ to below 5 but above 1. Since it is also a scalene triangle, the only value of $AC$ is 4. By adding 2, 3 and 4 together, you get an answer of 9.

The longest side of a triangle is less than the sum of its 2 other side. In this case, the longest possible side = 4, so perimeter = 2+3+4=9