To determine which side is the hypotenuse in a right triangle and to find the value of \( x \), we can apply the Pythagorean theorem, which states:
\[ c^2 = a^2 + b^2 \]
where \( c \) is the length of the hypotenuse and \( a \) and \( b \) are the lengths of the other two sides.
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Identifying the hypotenuse: In a right triangle, the hypotenuse is always the longest side. To identify the hypotenuse among the sides labeled 11.9 km, 14.7 km, and \( x \), the longest side must be taken as the hypotenuse.
- 11.9 km
- 14.7 km
- \( x \)
Since 14.7 km is greater than 11.9 km, we can determine that the side labeled 14.7 km is the hypotenuse.
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Calculating the value of \( x \): With the hypotenuse identified, we can use the Pythagorean theorem. Assuming the sides are as follows:
- \( a = 11.9 \) km
- \( b = x \)
- \( c = 14.7 \) km
We set up the equation:
\[ (14.7)^2 = (11.9)^2 + x^2 \]
Calculating the squares:
\[ 216.09 = 141.61 + x^2 \]
Now, isolate \( x^2 \):
\[ x^2 = 216.09 - 141.61 \] \[ x^2 = 74.48 \]
Now, take the square root to find \( x \):
\[ x \approx \sqrt{74.48} \approx 8.6 \text{ km} \]
Final Answers:
- The side labeled 14.7 km is the hypotenuse.
- The value of \( x \) is approximately 8.6 km.