Question
What is the length of the missing leg? One leg is 5.3 m and the slope is 8.3 m. If necessary, round to the nearest tenth.
Answers
GPT-4o mini
To find the length of the missing leg in a right triangle when one leg and the hypotenuse (the slope) are known, we can use the Pythagorean theorem. The theorem states that:
\[
a^2 + b^2 = c^2
\]
where \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the legs. In this case, let’s denote:
- One leg \(a = 5.3 \, \text{m}\)
- Hypotenuse \(c = 8.3 \, \text{m}\)
- The missing leg \(b\) is what we want to find.
Rearranging the Pythagorean theorem gives us:
\[
b^2 = c^2 - a^2
\]
Now substitute the known values:
\[
b^2 = (8.3)^2 - (5.3)^2
\]
Calculating the squares:
\[
(8.3)^2 = 68.89
\]
\[
(5.3)^2 = 28.09
\]
Now, subtract the two:
\[
b^2 = 68.89 - 28.09 = 40.8
\]
Now, find \(b\) by taking the square root:
\[
b = \sqrt{40.8} \approx 6.4 \, \text{m}
\]
Therefore, the length of the missing leg is approximately **6.4 m** when rounded to the nearest tenth.
\[
a^2 + b^2 = c^2
\]
where \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the legs. In this case, let’s denote:
- One leg \(a = 5.3 \, \text{m}\)
- Hypotenuse \(c = 8.3 \, \text{m}\)
- The missing leg \(b\) is what we want to find.
Rearranging the Pythagorean theorem gives us:
\[
b^2 = c^2 - a^2
\]
Now substitute the known values:
\[
b^2 = (8.3)^2 - (5.3)^2
\]
Calculating the squares:
\[
(8.3)^2 = 68.89
\]
\[
(5.3)^2 = 28.09
\]
Now, subtract the two:
\[
b^2 = 68.89 - 28.09 = 40.8
\]
Now, find \(b\) by taking the square root:
\[
b = \sqrt{40.8} \approx 6.4 \, \text{m}
\]
Therefore, the length of the missing leg is approximately **6.4 m** when rounded to the nearest tenth.