Question
a right triangle is shown with leg 1 being 20, the hypotenuse is 30 however leg 2 is unknown
Find the unknown side length in the given triangle. Round to the nearest hundredth.
Find the unknown side length in the given triangle. Round to the nearest hundredth.
Answers
GPT-4o mini
To find the length of the unknown leg (let's call it \( b \)) in a right triangle with one leg (\( a \)) measuring 20 and the hypotenuse (\( c \)) measuring 30, we can use the Pythagorean theorem, which states:
\[
a^2 + b^2 = c^2
\]
Plugging in the known values:
\[
20^2 + b^2 = 30^2
\]
Calculating the squares:
\[
400 + b^2 = 900
\]
Now, subtract 400 from both sides:
\[
b^2 = 900 - 400
\]
\[
b^2 = 500
\]
To find \( b \), take the square root of both sides:
\[
b = \sqrt{500}
\]
We can simplify \( \sqrt{500} \):
\[
b = \sqrt{100 \times 5} = \sqrt{100} \times \sqrt{5} = 10\sqrt{5}
\]
Calculating \( \sqrt{5} \) (approximately 2.236):
\[
b \approx 10 \times 2.236 = 22.36
\]
Thus, rounding to the nearest hundredth, the length of the unknown leg is:
\[
\boxed{22.36}
\]
\[
a^2 + b^2 = c^2
\]
Plugging in the known values:
\[
20^2 + b^2 = 30^2
\]
Calculating the squares:
\[
400 + b^2 = 900
\]
Now, subtract 400 from both sides:
\[
b^2 = 900 - 400
\]
\[
b^2 = 500
\]
To find \( b \), take the square root of both sides:
\[
b = \sqrt{500}
\]
We can simplify \( \sqrt{500} \):
\[
b = \sqrt{100 \times 5} = \sqrt{100} \times \sqrt{5} = 10\sqrt{5}
\]
Calculating \( \sqrt{5} \) (approximately 2.236):
\[
b \approx 10 \times 2.236 = 22.36
\]
Thus, rounding to the nearest hundredth, the length of the unknown leg is:
\[
\boxed{22.36}
\]