Question

1. what is the longest chord in a circle? a. the diameter b. a tangent line c. any chord d. a chord perpendicular to a radius 12. which equation represents a circle passing through the point (4,3) with a center at (2,1)? a. (x - 2)^2+(y - 1)^2 = 13 b. (x - 2)^2+(y - 1)^2 = 25 c. (x - 2)^2+(y - 1)^2 = 4 d. (x - 2)^2+(y - 1)^2 = 8 13. which property can be used to ensure that two tangent segments drawn from the same external point are equal? a. inscribed angle theorem b. alternate segment theorem c. pythagorean theorem d. congruence theorem

Explanation:

Step1: Recall chord definition

A chord is a line - segment joining two points on a circle. The diameter passes through the center of the circle and is the longest distance between two points on the circle among all chords. So for question 11, the answer is the diameter.

Step2: Use circle - equation formula

The standard form of a circle equation is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius. Given the center $(2,1)$ and a point $(4,3)$ on the circle, we find the radius $r$ using the distance formula $r=\sqrt{(4 - 2)^2+(3 - 1)^2}=\sqrt{4 + 4}=\sqrt{8}$. Then $r^2 = 8$, so the circle equation is $(x - 2)^2+(y - 1)^2=8$ for question 12.

Step3: Recall tangent - segment property

The property that two tangent segments drawn from the same external point to a circle are equal is based on congruence. We can prove the two right - triangles formed by the radius, tangent, and the line from the external point to the center of the circle are congruent. So for question 13, the relevant property is the congruence theorem.

Answer:

1. a. The diameter
2. d. $(x - 2)^2+(y - 1)^2=8$
3. d. Congruence Theorem