Question
On the unit circle, a right triangle with an angle measure of θ
has a leg opposite of θ
with a length of 12
what is the length of the other leg? Use the Pythagorean Theorem
3
3
2
2
3√2
the fraction with numerator square root of 3 and denominator 2
12
1 half
has a leg opposite of θ
with a length of 12
what is the length of the other leg? Use the Pythagorean Theorem
3
3
2
2
3√2
the fraction with numerator square root of 3 and denominator 2
12
1 half
Answers
GPT-4o mini
To find the length of the other leg of the right triangle using the Pythagorean Theorem, we start by using the relationship:
\[
a^2 + b^2 = c^2
\]
where:
- \( a \) is the length of one leg (12 in this case),
- \( b \) is the length of the other leg,
- \( c \) is the length of the hypotenuse.
Since the triangle is inscribed in the unit circle, the length of the hypotenuse \( c \) is 1 (the radius of the circle).
Given:
- \( a = 12 \)
- \( c = 1 \)
Plugging into the Pythagorean theorem:
\[
12^2 + b^2 = 1^2
\]
\[
144 + b^2 = 1
\]
\[
b^2 = 1 - 144
\]
\[
b^2 = -143
\]
Since \( b^2 = -143 \) leads to an invalid solution in this context (you cannot have a leg with a negative square), it indicates that having a leg of length 12 in a right triangle inscribed in a unit circle is impossible, because the maximum leg length cannot exceed the radius (which is 1).
Thus, the configuration is mathematically infeasible for the conditions given. If you have any other specific conditions or if the leg length is different, let me know!
\[
a^2 + b^2 = c^2
\]
where:
- \( a \) is the length of one leg (12 in this case),
- \( b \) is the length of the other leg,
- \( c \) is the length of the hypotenuse.
Since the triangle is inscribed in the unit circle, the length of the hypotenuse \( c \) is 1 (the radius of the circle).
Given:
- \( a = 12 \)
- \( c = 1 \)
Plugging into the Pythagorean theorem:
\[
12^2 + b^2 = 1^2
\]
\[
144 + b^2 = 1
\]
\[
b^2 = 1 - 144
\]
\[
b^2 = -143
\]
Since \( b^2 = -143 \) leads to an invalid solution in this context (you cannot have a leg with a negative square), it indicates that having a leg of length 12 in a right triangle inscribed in a unit circle is impossible, because the maximum leg length cannot exceed the radius (which is 1).
Thus, the configuration is mathematically infeasible for the conditions given. If you have any other specific conditions or if the leg length is different, let me know!