Question

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Sam is drawing triangles on banners using a ruler and protractor.

Fill in the Blank Question
On one banner Sam wants to create triangles with side lengths of 9 inches and 2 inches.

How many unique (one and only one) triangles with whole number side lengths can he make?

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Answer 1

To determine how many unique triangles can be formed with sides of lengths 9 inches and 2 inches, we need to use the triangle inequality theorem, which states that for any triangle with sides lengths \(a\), \(b\), and \(c\):

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

In this case, we have two sides: \(a = 9\) inches and \(b = 2\) inches. Let's denote the unknown side as \(c\).

Now, we apply the triangle inequality to find the permissible values of \(c\).

1. \(9 + 2 > c\) → \(c < 11\)
2. \(9 + c > 2\) → \(c > -7\) (This condition is always satisfied since \(c\) is a positive length)
3. \(2 + c > 9\) → \(c > 7\)

From these inequalities, we can combine the two relevant ones:
- From \(c < 11\)
- From \(c > 7\)

Thus, we have:
\[
7 < c < 11
\]

The whole number values for \(c\) that satisfy this inequality are:
- \(c = 8\)
- \(c = 9\)
- \(c = 10\)

Therefore, there are 3 unique lengths for the third side that will create valid triangles with the given two sides of lengths 9 inches and 2 inches.

The answer is:
\[
\boxed{3}
\]