Question
What is the length of the missing leg? one leg is 5 mi and the slope is 10 mi. If necessary, round to the nearest tenth.
Answers
GPT-4o mini
To find the length of the missing leg in a right triangle, we can use the Pythagorean theorem, which states:
\[ a^2 + b^2 = c^2 \]
where \( a \) and \( b \) are the lengths of the legs, and \( c \) is the length of the hypotenuse (the slope in this case).
In your problem:
- One leg \( a = 5 \) mi
- The hypotenuse \( c = 10 \) mi
- We need to find the other leg \( b \).
Substituting the values into the equation:
\[ 5^2 + b^2 = 10^2 \]
\[ 25 + b^2 = 100 \]
Now, subtract 25 from both sides:
\[ b^2 = 100 - 25 \]
\[ b^2 = 75 \]
Now, take the square root of both sides to find \( b \):
\[ b = \sqrt{75} \]
To simplify \( \sqrt{75} \):
\[ \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} \]
Now, calculating \( \sqrt{3} \) (approximately 1.732):
\[ b \approx 5 \times 1.732 \approx 8.66 \]
Rounding to the nearest tenth:
\[ b \approx 8.7 \]
So, the length of the missing leg is approximately **8.7 miles**.
\[ a^2 + b^2 = c^2 \]
where \( a \) and \( b \) are the lengths of the legs, and \( c \) is the length of the hypotenuse (the slope in this case).
In your problem:
- One leg \( a = 5 \) mi
- The hypotenuse \( c = 10 \) mi
- We need to find the other leg \( b \).
Substituting the values into the equation:
\[ 5^2 + b^2 = 10^2 \]
\[ 25 + b^2 = 100 \]
Now, subtract 25 from both sides:
\[ b^2 = 100 - 25 \]
\[ b^2 = 75 \]
Now, take the square root of both sides to find \( b \):
\[ b = \sqrt{75} \]
To simplify \( \sqrt{75} \):
\[ \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} \]
Now, calculating \( \sqrt{3} \) (approximately 1.732):
\[ b \approx 5 \times 1.732 \approx 8.66 \]
Rounding to the nearest tenth:
\[ b \approx 8.7 \]
So, the length of the missing leg is approximately **8.7 miles**.