Question

1. what is the radius of the circle with the equation (x - 1)^2+(y - 4)^2 = 49? a. 5 b. 4 c. 6 d. 7 15. if two chords are equidistant from the center of a circle, what can you conclude about their lengths? a. they are congruent. b. one is half as long as the other. c. one is twice as long as the other. d. they are perpendicular to each other. 16. what is the relationship between the radius and the tangent line at the point of tangency? a. they are parallel b. they are collinear c. they are equal in length d. they are perpendicular

Explanation:

Step1: Recall circle - equation formula

The standard form of a circle's equation is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius.

Step2: Identify radius from given equation

For the equation $(x - 1)^2+(y - 4)^2 = 49$, we have $r^2=49$. Taking the square - root of both sides, $r=\sqrt{49}=7$.

Step3: Recall chord - distance property

If two chords are equidistant from the center of a circle, then they are congruent. This is a property of circles based on the symmetry and the relationship between the perpendicular distance from the center to the chords and their lengths.

Step4: Recall radius - tangent relationship

The radius of a circle and the tangent line at the point of tangency are perpendicular. This is a fundamental property of circles, which can be proven using geometric principles such as the fact that the shortest distance from the center of the circle to the tangent is along the radius.

Answer:

1. d. 7
2. a. They are congruent.
3. d. They are perpendicular.