In a 45-45-90 triangle, the angles are 45 degrees, 45 degrees, and 90 degrees, which means the two legs of the triangle are equal in length. If we label the lengths of the legs as \( r \), then by the properties of a 45-45-90 triangle, the coordinates of the point where the triangle intersects the unit circle can be determined.
The unit circle is defined by the equation \( x^2 + y^2 = 1 \). We know that in a 45-45-90 triangle, the coordinates can be expressed as \( (r, r) \) because the legs of the triangle are equal.
Since the line \( r = 1 \) indicates that the hypotenuse of the triangle (which is also the radius of the unit circle) has a length of 1, the point of intersection where the triangle meets the circle has coordinates \( (r, r) \) where \( r = \frac{1}{\sqrt{2}} \) due to the triangle's properties.
To find the x-coordinate of this point of intersection:
- We calculate \( \frac{1}{\sqrt{2}} \).
- The x-coordinate is \( x = \frac{1}{\sqrt{2}} \).
This value can be simplified further: \[ \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]
Thus, the x-coordinate of the point where the 45-45-90 triangle intersects the unit circle is \( \frac{\sqrt{2}}{2} \). However, for the purpose of this question, since the triangle is also intersecting the unit circle, we can simply use the value of 1 at 45 degrees.
So the answer is:
The x-coordinate is 1.