Question

Two sides of a triangle have lengths 3 cm and 8 cm respectively. The third side has a length of x cm, where x is an integer. Write down an inequality that must be satisfied by x

Answer 1

Answer: 5 < x < 11

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Step-by-step explanation:

Let's label the sides of the triangle a, b, and c.

Furthermore, let's have 'a' & b represent known sides and
`b \ge a` be the case. For a triangle to be possible, the missing side c must have these restrictions placed on it:

b-a < c < b+a

This inequality is valid because of a modification to the Triangle Inequality Theorem. I'll leave the proof to the reader.

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In this case we have a = 3 and b = 8 which lead to...

b-a < c < b+a

8-3 < x < 8+3

5 < x < 11

The third side is between 5 and 11, excluding both endpoints. We cannot have x = 5. Furthermore, we cannot have x = 11 either. If x were either of those endpoints, then we'd form a straight line rather than a triangle.

Having x be an integer leads to the roster set notation {6,7,8,9,10} to represent all possible values of x. For instance, we could have a triangle with side lengths 3, 8, and 6. I recommend getting slips of paper of these lengths to try it out yourself. Or you can use software such as GeoGebra to see why this works.