Question
Find the unknown side length in the given triangle. Round to the nearest hundredth.(1 point) alt="A right triangle shows the base without any value, the hypotenuse on the right with a value of 30, and the perpendicular side to the base on the left with a value of 20."
Answers
GPT-4o mini
To find the unknown side length in the right triangle, we can use the Pythagorean theorem, which states:
\[ a^2 + b^2 = c^2 \]
where:
- \( a \) and \( b \) are the lengths of the two legs (sides) of the triangle,
- \( c \) is the length of the hypotenuse.
In this case, we have:
- Hypotenuse \( c = 30 \)
- One leg \( a = 20 \)
- We need to find the length of the other leg \( b \).
Plugging the values into the Pythagorean theorem:
\[
20^2 + b^2 = 30^2
\]
Calculating \( 20^2 \) and \( 30^2 \):
\[
400 + b^2 = 900
\]
Now, subtract 400 from both sides:
\[
b^2 = 900 - 400
\]
\[
b^2 = 500
\]
Now, take the square root of both sides to find \( b \):
\[
b = \sqrt{500}
\]
Calculating \( \sqrt{500} \):
\[
b \approx 22.36
\]
Rounding to the nearest hundredth, we find that the unknown side length is approximately:
\[
\boxed{22.36}
\]
\[ a^2 + b^2 = c^2 \]
where:
- \( a \) and \( b \) are the lengths of the two legs (sides) of the triangle,
- \( c \) is the length of the hypotenuse.
In this case, we have:
- Hypotenuse \( c = 30 \)
- One leg \( a = 20 \)
- We need to find the length of the other leg \( b \).
Plugging the values into the Pythagorean theorem:
\[
20^2 + b^2 = 30^2
\]
Calculating \( 20^2 \) and \( 30^2 \):
\[
400 + b^2 = 900
\]
Now, subtract 400 from both sides:
\[
b^2 = 900 - 400
\]
\[
b^2 = 500
\]
Now, take the square root of both sides to find \( b \):
\[
b = \sqrt{500}
\]
Calculating \( \sqrt{500} \):
\[
b \approx 22.36
\]
Rounding to the nearest hundredth, we find that the unknown side length is approximately:
\[
\boxed{22.36}
\]