Question

select the correct answer.
how does the figure help verify the triangle inequality theorem?
figure with a line segment labeled 15, two triangles with sides 7 and 4 respectively
a. by showing that only one triangle can be formed when the sum of the lengths of two sides equals the length of the third
b. 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
c. by showing that a triangle cannot be formed when the sum of the lengths of two sides equals the length of the third
d. 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
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Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that for a 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 remaining side. Mathematically, \(a + b>c\), \(a + c>b\), and \(b + c>a\). If the sum of two sides is less than or equal to the third side, a triangle cannot be formed.

Step2: Analyze the Given Figure and Options

• For the given lengths (7, 4, and 15), let's check the sum of 7 and 4: \(7 + 4=11\), and \(11<15\).
• Option A: Says a triangle can be formed when sum of two sides equals the third. But if \(a + b = c\), the two sides would lie along the third side, not forming a triangle. So A is incorrect.
• Option B: Says a triangle can be formed when sum of two sides is less than the third. But by triangle inequality, if \(a + b
• Option C: Says a triangle cannot be formed when sum of two sides equals the third. The figure here has \(7 + 4 = 11

eq15\), so this is not the case here. So C is incorrect.

• Option D: Says a triangle cannot be formed when sum of two sides is less than the third. Since \(7+4 = 11<15\), the figure shows that a triangle cannot be formed in this case, which verifies the triangle inequality theorem (that a triangle cannot be formed when the sum of two sides is less than the third side).

Answer:

D. 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