What is the cosine of an angle expressed as a rationalized fraction for the point (-2, 7)?

To find the cosine of the angle corresponding to the point (-2, 7), we first need to determine the radius (or distance from the origin) for that point in the Cartesian plane. The distance from the origin (0, 0) to the point (-2, 7) can be found using the distance formula:

[

r = \sqrt{x^2 + y^2}

]

Substituting the coordinates of the point:

[

r = \sqrt{(-2)^2 + 7^2} = \sqrt{4 + 49} = \sqrt{53}

]

Next, cosine is defined as the adjacent side over the hypotenuse in a right triangle formed by these coordinates. In this case, the x-coordinate represents the adjacent side while the hypotenuse is the distance we just calculated:

[

\cos(\theta) = \frac{x}{r} = \frac{-2}{\sqrt{53}}

]

To express this value as a rationalized fraction, we multiply the numerator and the denominator by (\sqrt{53}):

[

\cos(\theta) = \frac{-2\sqrt{53}}{53}

]

This gives us the