Question

in △xyz, xy = 13√2 m and yz = 5√2 m. which lengths are possible for xz? select all correct answers. a. 11.5 m b. 13 m c. 18.5 m d. 26 m

Explanation:

Step1: Recall Triangle Inequality Theorem

For a triangle with sides \(a\), \(b\), \(c\), the following must hold:
\( |a - b| < c < a + b \)

Here, \( XY = 13\sqrt{2} \, \text{m} \) and \( YZ = 5\sqrt{2} \, \text{m} \). Let \( XZ = c \), \( a = 13\sqrt{2} \), \( b = 5\sqrt{2} \).

Step2: Calculate the range for \( XZ \)

First, find \( a - b \):
\( 13\sqrt{2} - 5\sqrt{2} = 8\sqrt{2} \)
Calculate the numerical value: \( 8\sqrt{2} \approx 8 \times 1.414 = 11.312 \)

Next, find \( a + b \):
\( 13\sqrt{2} + 5\sqrt{2} = 18\sqrt{2} \)
Calculate the numerical value: \( 18\sqrt{2} \approx 18 \times 1.414 = 25.452 \)

So, the length of \( XZ \) must satisfy \( 11.312 < XZ < 25.452 \)

Step3: Check each option

• Option A: \( 11.5 \, \text{m} \)

\( 11.312 < 11.5 < 25.452 \), so this is possible.

• Option B: \( 13 \, \text{m} \)

\( 11.312 < 13 < 25.452 \), so this is possible.

• Option C: \( 18.5 \, \text{m} \)

\( 11.312 < 18.5 < 25.452 \), so this is possible.

• Option D: \( 26 \, \text{m} \)

\( 26 > 25.452 \), so this is not possible.

Answer:

A. 11.5 m, B. 13 m, C. 18.5 m