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One of the legs of a right triangle measures 6 cm and its hypotenuse measures 9 cm. Find the measure of the other leg. If neces...Asked by savannah
One of the legs of a right triangle measures 5 cm and its hypotenuse measures 10 cm. Find the measure of the other leg. If necessary, round to the nearest tenth.
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Answered by
GPT-4o mini
To find the measure of the other leg of the right triangle, we can use the Pythagorean theorem, which 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, we know one leg \(a = 5\) cm and the hypotenuse \(c = 10\) cm. We need to find the length of the other leg \(b\).
Substituting the known values into the Pythagorean theorem gives us:
\[
5^2 + b^2 = 10^2
\]
Calculating the squares:
\[
25 + b^2 = 100
\]
Next, subtract 25 from both sides:
\[
b^2 = 100 - 25
\]
\[
b^2 = 75
\]
Now, take the square root of both sides to find \(b\):
\[
b = \sqrt{75}
\]
We can simplify \(\sqrt{75}\):
\[
b = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}
\]
Now, we can approximate \(b\) using the value of \(\sqrt{3} \approx 1.732\):
\[
b \approx 5 \times 1.732 \approx 8.66
\]
Rounding to the nearest tenth, we find:
\[
b \approx 8.7 \text{ cm}
\]
Thus, the measure of the other leg is approximately **8.7 cm**.
\[
a^2 + b^2 = c^2
\]
where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse. In this case, we know one leg \(a = 5\) cm and the hypotenuse \(c = 10\) cm. We need to find the length of the other leg \(b\).
Substituting the known values into the Pythagorean theorem gives us:
\[
5^2 + b^2 = 10^2
\]
Calculating the squares:
\[
25 + b^2 = 100
\]
Next, subtract 25 from both sides:
\[
b^2 = 100 - 25
\]
\[
b^2 = 75
\]
Now, take the square root of both sides to find \(b\):
\[
b = \sqrt{75}
\]
We can simplify \(\sqrt{75}\):
\[
b = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}
\]
Now, we can approximate \(b\) using the value of \(\sqrt{3} \approx 1.732\):
\[
b \approx 5 \times 1.732 \approx 8.66
\]
Rounding to the nearest tenth, we find:
\[
b \approx 8.7 \text{ cm}
\]
Thus, the measure of the other leg is approximately **8.7 cm**.
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