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.(1 point) Responses 2 2 12 1 half 3 3 3√2 the fraction with numerator square root of 3 and denominator 2 Skip to navigation

Answer 1

To determine the length of the other leg of the triangle given that the leg opposite to angle \( \theta \) has a length of 12, we can use the Pythagorean Theorem.

The Pythagorean Theorem states:

\[ a^2 + b^2 = c^2 \]

where \( c \) is the hypotenuse, and \( a \) and \( b \) are the lengths of the legs of the triangle. In this case, we have:

• Leg opposite \( \theta \) (let's call it \( a \)): 12
• The other leg (let's call it \( b \)): unknown
• The hypotenuse (let's call it \( c \)): since the right triangle is inscribed in the unit circle, \( c = 1 \) (the length of the radius of the unit circle).

Plugging in the known values, we get:

\[ 12^2 + b^2 = 1^2 \]

Calculating \( 12^2 \):

\[ 144 + b^2 = 1 \]

Now, rearranging to solve for \( b^2 \):

\[ b^2 = 1 - 144 \] \[ b^2 = -143 \]

Since \( b^2 \) cannot be negative in the context of real numbers, it indicates that a leg of length 12 cannot exist in a right triangle inscribed in the unit circle (where the hypotenuse must be 1). Therefore, the given length of the opposite leg (12) exceeds the hypotenuse (1) in a valid right triangle on the unit circle.

Thus, under these constraints, no valid length for the other leg exists.

In conclusion, the provided information leads to a contradiction, and the other leg cannot be computed as specified.

Answer 2

since you are given a side and an angle, the Pythagorean Theorem is of no use.
Instead, if the other leg has length x, then
tanθ = 12/x, so x = 12 cotθ

Answer 3

You're correct! Thank you for pointing that out. Since you were given a triangle formed with an angle \( \theta \) and one leg (the opposite leg) having a length of 12, we can indeed use trigonometric relationships instead of the Pythagorean Theorem.

Given:

• Opposite side: \( 12 \)
• Let the other leg be \( x \) (the adjacent side).

Using the tangent function, we have:

\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{x} \]

From this, we can express \( x \) in terms of \( \theta \):

\[ x = \frac{12}{\tan \theta} \]

Alternatively, using the cotangent function:

\[ \tan \theta = \frac{12}{x} \implies x = 12 \cot \theta \]

This approach directly relates the lengths of the triangle's sides to the angle \( \theta \).

The answer to your question about the length of the other leg \( x \) is therefore:

\[ x = 12 \cot \theta \]

If specific values for \( \theta \) were provided, you could calculate \( x \) using that formula. But without a numerical value for \( \theta \), this is the correct expression for the length of the other leg.