Question

select the correct answer.

how does the figure help verify the triangle inequality theorem?

by showing that only one triangle can be formed when the sum of the lengths of two sides is less than the length of the third
by showing that a triangle cannot be formed when the sum of the lengths of two sides is less than the length of the third side
by showing that only one triangle can be formed when the sum of the lengths of two sides equals the length of the third
by showing that a triangle cannot be formed when the sum of the lengths of two sides equals the length of the third

Explanation:

Identify the given values

Using the Triangle Inequality Theorem knowledge point

$$
LATEXBLOCK0
$$

Compare the side lengths

Using the Triangle Inequality Theorem knowledge point

$$
LATEXBLOCK1
$$

Interpret the geometric representation

Using the Triangle Inequality Theorem knowledge point

• The arcs represent all possible locations for the free endpoints of the sides of lengths \(7\) and \(4\).
• Since the sum of their lengths (\(11\)) is less than the base length (\(15\)), the arcs do not intersect.
• This demonstrates that a triangle cannot be formed because the sum of the lengths of two sides is less than the length of the third side.

Answer:

• by showing that only one triangle can be formed when the sum of the lengths of two sides is less than the length of the third
• by showing that a triangle cannot be formed when the sum of the lengths of two sides is less than the length of the third side (Correct answer)
• by showing that only one triangle can be formed when the sum of the lengths of two sides equals the length of the third
• by showing that a triangle cannot be formed when the sum of the lengths of two sides equals the length of the third