Question
What is the length of the missing leg? One leg is 6 mm and the slope is 9 mm. If necessary, round to the nearest tenth.
Answers
GPT-4o mini
To find the length of the missing leg in a right triangle where one leg is given and the hypotenuse (slope) is known, we can use the Pythagorean theorem. The theorem states that:
\[
a^2 + b^2 = c^2
\]
where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse.
In this case:
- One leg (\(a\)) = 6 mm
- Hypotenuse (\(c\)) = 9 mm
- Let's find the other leg (\(b\)).
Rearranging the formula gives:
\[
b^2 = c^2 - a^2
\]
Substituting the known values:
\[
b^2 = 9^2 - 6^2
\]
\[
b^2 = 81 - 36
\]
\[
b^2 = 45
\]
Now, we take the square root to find \(b\):
\[
b = \sqrt{45}
\]
Simplifying \(\sqrt{45}\):
\[
b = \sqrt{9 \cdot 5} = 3\sqrt{5}
\]
Calculating \(3\sqrt{5}\):
Using the approximate value of \(\sqrt{5} \approx 2.236\):
\[
b \approx 3 \times 2.236 \approx 6.708
\]
Rounding this to the nearest tenth:
\[
b \approx 6.7 \text{ mm}
\]
Thus, the length of the missing leg is approximately **6.7 mm**.
\[
a^2 + b^2 = c^2
\]
where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse.
In this case:
- One leg (\(a\)) = 6 mm
- Hypotenuse (\(c\)) = 9 mm
- Let's find the other leg (\(b\)).
Rearranging the formula gives:
\[
b^2 = c^2 - a^2
\]
Substituting the known values:
\[
b^2 = 9^2 - 6^2
\]
\[
b^2 = 81 - 36
\]
\[
b^2 = 45
\]
Now, we take the square root to find \(b\):
\[
b = \sqrt{45}
\]
Simplifying \(\sqrt{45}\):
\[
b = \sqrt{9 \cdot 5} = 3\sqrt{5}
\]
Calculating \(3\sqrt{5}\):
Using the approximate value of \(\sqrt{5} \approx 2.236\):
\[
b \approx 3 \times 2.236 \approx 6.708
\]
Rounding this to the nearest tenth:
\[
b \approx 6.7 \text{ mm}
\]
Thus, the length of the missing leg is approximately **6.7 mm**.