Question
if it is a right traingle and one side is 10 miles and the bottom is 5 miles what is the last side?
Answers
GPT-4o mini
To find the length of the third side of a right triangle when two sides are known, we can use the Pythagorean theorem. The theorem states that:
\[ a^2 + b^2 = c^2 \]
where \( c \) is the hypotenuse (the longest side), and \( a \) and \( b \) are the lengths of the other two sides.
In your case, you have one side of 10 miles and another side of 5 miles. Assuming that the side of 10 miles is the hypotenuse, we can set it up as follows:
1. Let \( a = 5 \) miles
2. Let \( c = 10 \) miles
Now we need to find \( b \):
\[ a^2 + b^2 = c^2 \]
\[ 5^2 + b^2 = 10^2 \]
\[ 25 + b^2 = 100 \]
\[ b^2 = 100 - 25 \]
\[ b^2 = 75 \]
\[ b = \sqrt{75} \]
\[ b = 5\sqrt{3} \text{ miles} \approx 8.66 \text{ miles} \]
So, the length of the third side \( b \) is approximately 8.66 miles.
If instead, the 10 miles side is one of the legs (not the hypotenuse), we can solve for the hypotenuse:
Let:
- \( a = 5 \) miles
- \( b = 10 \) miles
Then the calculation would be:
\[ a^2 + b^2 = c^2 \]
\[ 5^2 + 10^2 = c^2 \]
\[ 25 + 100 = c^2 \]
\[ 125 = c^2 \]
\[ c = \sqrt{125} \]
\[ c = 5\sqrt{5} \text{ miles} \approx 11.18 \text{ miles} \]
So, depending on which side is the hypotenuse, the third side could either be approximately 8.66 miles (if 10 miles is the hypotenuse) or approximately 11.18 miles (if 10 miles is one of the legs).
\[ a^2 + b^2 = c^2 \]
where \( c \) is the hypotenuse (the longest side), and \( a \) and \( b \) are the lengths of the other two sides.
In your case, you have one side of 10 miles and another side of 5 miles. Assuming that the side of 10 miles is the hypotenuse, we can set it up as follows:
1. Let \( a = 5 \) miles
2. Let \( c = 10 \) miles
Now we need to find \( b \):
\[ a^2 + b^2 = c^2 \]
\[ 5^2 + b^2 = 10^2 \]
\[ 25 + b^2 = 100 \]
\[ b^2 = 100 - 25 \]
\[ b^2 = 75 \]
\[ b = \sqrt{75} \]
\[ b = 5\sqrt{3} \text{ miles} \approx 8.66 \text{ miles} \]
So, the length of the third side \( b \) is approximately 8.66 miles.
If instead, the 10 miles side is one of the legs (not the hypotenuse), we can solve for the hypotenuse:
Let:
- \( a = 5 \) miles
- \( b = 10 \) miles
Then the calculation would be:
\[ a^2 + b^2 = c^2 \]
\[ 5^2 + 10^2 = c^2 \]
\[ 25 + 100 = c^2 \]
\[ 125 = c^2 \]
\[ c = \sqrt{125} \]
\[ c = 5\sqrt{5} \text{ miles} \approx 11.18 \text{ miles} \]
So, depending on which side is the hypotenuse, the third side could either be approximately 8.66 miles (if 10 miles is the hypotenuse) or approximately 11.18 miles (if 10 miles is one of the legs).