A triangle has no congruent sides. The lengths of two sides of the triangle are AB and BD . Use the number line above to determine the lengths of AB and BD . Then, use the Triangle Inequality Theorem to determine which two segments have lengths that could be used as the third side of the triangle
Without a specific number line provided, we cannot determine the lengths of AB and BD. However, we can still determine the possible lengths for the third side of the triangle using the Triangle Inequality Theorem.
The Triangle Inequality Theorem states that for any triangle with side lengths a, b, and c, the sum of the lengths of any two sides must be greater than the length of the third side.
In this case, let's assume that AB and BD are the two sides of the triangle, and we are trying to determine the possible lengths for the third side.
By applying the Triangle Inequality Theorem, we can write the following inequalities:
AB + BD > third side
AB + third side > BD
BD + third side > AB
Now, by substituting AB and BD, we can rewrite the inequalities:
2AB > third side
AB + third side > BD
BD + third side > AB
These inequalities hold true as long as the third side is greater than the difference between AB and BD, but less than the sum of AB and BD.
To summarize, the possible lengths for the third side of the triangle are any values greater than the absolute value of the difference between AB and BD, but less than the sum of AB and BD.