Two sides of a triangle are 10 mm and 7 mm. Using the Triangle Inequality Theorem (see picture above), determine the length of the third side from the choices given.

Question

(1 point)
Responses

3 mm
3 mm

5 mm
5 mm

1 mm
1 mm

2 mm

Answer 1

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Given the two sides of the triangle are 10 mm and 7 mm, we can denote the third side as \( x \). According to the Triangle Inequality Theorem, we have:

1. \( 10 + 7 > x \) → \( x < 17 \)
2. \( 10 + x > 7 \) → \( x > -3 \) (always true in this context)
3. \( 7 + x > 10 \) → \( x > 3 \)

From the inequalities derived:

• The third side \( x \) must be less than 17.
• The third side \( x \) must be greater than 3.

Combining these results, we find that \( 3 < x < 17 \).

Now, looking at the given choices for the length of the third side:

• 3 mm (not valid since \( x \) must be greater than 3)
• 5 mm (valid as it lies within the range)
• 1 mm (not valid since \( x \) must be greater than 3)
• 2 mm (not valid since \( x \) must be greater than 3)

The only length that satisfies these conditions is:

5 mm