A special right triangle drawn in Quadrant I intersects the unit circle at point (x,y)%0D%0A(%0D%0A𝑥%0D%0A,%0D%0A𝑦%0D%0A)%0D%0A where y=2√2%0D%0A𝑦%0D%0A=%0D%0A2%0D%0A2%0D%0A. What is the value of x in point (x,y)%0D%0A(%0D%0A𝑥%0D%0A,%0D%0A𝑦%0D%0A)%0D%0A? Use the equation of the unit circle to determine the missing coordinate.(1 point)

1 answer

To find the value of \( x \) for the point \( (x, y) \) where \( y = 2\sqrt{2} \) and the point lies on the unit circle, we start with the equation of the unit circle:

\[ x^2 + y^2 = 1 \]

Given \( y = 2\sqrt{2} \), we can substitute this value into the unit circle equation:

\[ x^2 + (2\sqrt{2})^2 = 1 \]

Calculating \( (2\sqrt{2})^2 \):

\[ (2\sqrt{2})^2 = 4 \cdot 2 = 8 \]

Now substitute this back into the equation:

\[ x^2 + 8 = 1 \]

Next, we isolate \( x^2 \):

\[ x^2 = 1 - 8 \] \[ x^2 = -7 \]

Since \( x^2 = -7 \) does not yield a real number (as \( x^2 \) cannot be negative in the real number system), this indicates that the value of \( y = 2\sqrt{2} \) does not correspond to a point on the unit circle in Quadrant I. The maximum value for \( y \) on the unit circle is \( 1 \), thus \( 2\sqrt{2} \) exceeds that limit.

Therefore, there is no valid value for \( x \) corresponding to \( y = 2\sqrt{2} \) on the unit circle.