Question

the locations of three friends homes for the triangle shown.
what is the possible range for the distances between emmas house and claudias house?
\\( x > 4 \\)
\\( x < 10 \\)
\\( 4 < x < 21 \\)
\\( 4 < x < 10 \\)

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that for any triangle, the length of any side must be greater than the absolute difference of the other two sides and less than the sum of the other two sides. Let the sides be \( a \), \( b \), and \( c \), then \( |a - b| < c < a + b \).

Step2: Identify the known sides

In this triangle, the two known sides are \( 7 \) miles (Claudia to Dane) and \( 3 \) miles (Emma to Dane). Let \( x \) be the distance between Emma and Claudia.

Step3: Apply the triangle inequality

First, find the difference of the two known sides: \( |7 - 3| = 4 \). Then, find the sum of the two known sides: \( 7 + 3 = 10 \). So, by the triangle inequality theorem, \( 4 < x < 10 \).

Answer:

\( 4 < x < 10 \) (corresponding to the option with this inequality)