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There are three or there are four, depending which definition you have been taught. If you are of the school that is taught that an isosceles triangle has 2 equal sides and 2 equal angles, there are three. If, however, you were taught that an isosceles triangle has at least 2 equal sides and at least 2 equal angles, then there are four: 5+5+8 6+6+6 Note: Strictly, this one is equalateral but conforms to an isosceles by the second theory. 7+7+4 8+8+2 The two equal sides, when added together, must be equal to or greater than the third side or they will not join up.

There would four different isosceles triangles that have perimeter of 18 and side lengths of natural numbers.

What is an isosceles triangle?

"It is triangle which has any two sides equal in length and angles opposite to equal sides are equal in measure."

What is triangle inequality?

"It states that the sum of any two sides of a triangle must be greater than the third side."

What is perimeter of triangle?

"It is the sum of all sides of an isosceles triangle."

For given question,

Let, a = the length of similar sides of an isosceles triangle

b = the length of the base of an isosceles triangle

Also, a and b are natural numbers.

Using the triangle inequality theorem we get:

⇒ a + a > b

⇒ 2a > b > 0                       .................(i)

Here, the perimeter of an isosceles triangle is 18.

⇒ 2a + b = 18

⇒ b = 18 - 2a                    ..................(ii)

We know, b > 0

⇒ 18 - 2a > 0

⇒ 18 > 2a

⇒ 9 > a                      .....................(iii)

Substitute b = 18 - 2a into (i),

⇒ 2a > 18 - 2a

⇒ 4a > 18

⇒ a > 18/4

⇒ a > 4                  .........................(iv)

From (iii) and (iv),

4 < a < 9

for a = 5,

⇒ b = 18 - 2 × 5

⇒ b = 8

for a = 6,

⇒ b = 18 - 2 × 6

⇒ b = 6

for a = 7,

⇒ b = 18 - 2 × 7

⇒ b = 3

for a = 8,

⇒ b = 18 - 2 × 8

⇒ b = 2

This means, there are four possible values of a and b.

Therefore, there would four different isosceles triangles that have perimeter of 18 and side lengths of natural numbers.

Learn more about the triangle inequality theorem here:

https://brainly.com/question/1163433

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