Question

Two sides of a triangle have the following measures. Find the range of possible measures for the third side.

6, 10

Question 20 options:

8<x<15


5<x<15


4<x<16


5<x<16

Answer 1

To find the range of possible measures for the third side of a triangle when the other two sides are 6 and 10, we can use the triangle inequality theorem. The theorem states that:

1. The sum of the lengths of any two sides must be greater than the length of the third side.

Let \( x \) be the length of the third side. We will set up the inequalities based on the triangle inequality:

1. \( 6 + 10 > x \)
- This gives us: \( x < 16 \)

2. \( 6 + x > 10 \)
- This simplifies to: \( x > 4 \)

3. \( 10 + x > 6 \)
- This simplifies to: \( x > -4 \) (which is always true since \( x \) is positive)

Now, combining the relevant inequalities from the first two conditions, we have:

\[
4 < x < 16
\]

Looking at the options provided:

- 8 < x < 15
- 5 < x < 15
- 4 < x < 16
- 5 < x < 16

The correct answer is:

**4 < x < 16**.