Question

Mack is drawing a triangle. She wants to use the side lengths of 15 in and 8 in for two of her sides.

Which of the lengths below could be the third side of Mack's triangle?


4 in

9 in

6 in

3 in

Answer 1

To determine which length could be the third side of Mack's triangle, we can apply the triangle inequality theorem. This theorem states that for any triangle with side lengths a, b, and c, the following inequalities must hold:

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

In this case, we have two sides of lengths 15 in and 8 in. Let’s denote the third side as \( x \).

Using the triangle inequalities, we can formulate the following conditions:

1. \( 15 + 8 > x \) → \( 23 > x \) (or \( x < 23 \))
2. \( 15 + x > 8 \) → \( x > 8 - 15 \) → \( x > -7 \) (this is always true for positive lengths)
3. \( 8 + x > 15 \) → \( x > 15 - 8 \) → \( x > 7 \)

From these inequalities, we find that the third side \( x \) must satisfy:

\[ 7 < x < 23 \]

Now, we can check which of the provided lengths (4 in, 9 in, 6 in, 3 in) fits this condition:

- 4 in: Not valid (4 ≤ 7)
- 9 in: Valid (7 < 9 < 23)
- 6 in: Not valid (6 ≤ 7)
- 3 in: Not valid (3 ≤ 7)

Thus, the only possible length for the third side of Mack's triangle is **9 in**.