Question

Two sides of a triangle measure 4 ft and 9 ft. Which length is possible for the third side of the triangle?

Multiple choice question.
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A)
2 ft

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B)
5 ft

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C)
12 ft

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D)
15 ft

Answer 1

To determine the possible lengths for the third side of a triangle when two sides are given, we can apply the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given two sides of a triangle measuring 4 ft and 9 ft, let \( x \) be the length of the third side. The triangle inequalities that must be satisfied are:

1. \( 4 + 9 > x \)  →   \( 13 > x \)   →   \( x < 13 \)
2. \( 4 + x > 9 \)  →   \( x > 5 \)
3. \( 9 + x > 4 \)  →   \( x > -5 \) (This inequality is always true since \( x \) must be a positive length.)

From the inequalities, we combine \( x > 5 \) and \( x < 13 \). Therefore, the length of the third side \( x \) must satisfy:

\[ 5 < x < 13 \]

Now let's evaluate the choices:

A) 2 ft - Not possible (2 < 5)
B) 5 ft - Not possible (5 is not greater than 5)
C) 12 ft - Possible (5 < 12 < 13)
D) 15 ft - Not possible (15 > 13)

The only length that is possible for the third side of the triangle is:

**C) 12 ft**