Question

Triangle A B C is shown. Side A B has a length of 12. Side B C has a length of x. Side A C has a length of 15. The value of x must be greater than ________. 0 1 3 7

Answer 1

Here we have to use the Triangle Inequality Theorem, because the problem it's asking about the restriction of one side, with two sides given. This theorem, in words, states that the sum of the minor sides is more than the length of the major sides of all three. So, if we translate this to equation, we have:

`AB + BC > AC`

Where
`AB=12`;
`BC=x` and
`AC=15`

Assuming that
`AC` is the longest side we applied the theorem to this problem. Now, replacing all values we have:

`12+x>15\\x>15-12\\x>3`

Therefore, the unknown side must be greater than 3 to not violate the Triangle Inequality Theorem, which is always true about every triangle.

Answer 2

3

Explanation:

According to the Triangle Inequality Theorem, the sum of any 2 sides of any triangle must be greater than the length of the third side. That is:

AB + BC > AC and AB + AC > BC and AC + BC > AB

So in this case:

if AC + AB > BC

x + 12 > 15

After substracting 12 from each side of the equation, we get that:

x > 3

Hence, the value of x must be greater than 3.