15 which of the following represents a circle with its center at the origin and a radius of 10?
a. $x^2 + y^2 = 100$
b. $x^2 + y^2 = 10$
c. $(x + 10)^2 + (y + 10)^2 = 100$
d. $x^2 - y^2 = 100$
16 what is the center of the circle given by $(x + 3)^2 + (y - 4)^2 = 49$?
a. $(3, -4)$
b. $(3, 4)$
c. $(-3, 4)$
d. $(-3, -4)$
17 what does the alternate segment theorem state regarding tangents and circles?
a. the tangent is parallel to the chord
b. the tangent and chord are equal in length
c. the alternate segment intersects the circle
d. the angle between the tangent and chord is equal to the angle in the alternate segment
18 which property can be used to ensure that two tangent segments drawn from the same external point are equal?
a. alternate segment theorem
b. congruence theorem
c. pythagorean theorem
d. inscribed angle theorem
19 which of the following is a property of the radian measure?
a. radians are primarily used for measuring volume
b. radians directly relate the arc length to the radius
c. radians measure the area of a sector
d. radians are independent of the circle’s radius
20 if a circle has a radius of 7 meters and a central angle of $60^circ$, what is the length of the arc?
a. $\frac{7}{3}pi$ meters
b. $\frac{3}{7}$ meters
c. $14pi$ meters
d. 7 meters
21 which of the following points lies on the circle with equation $(x + 3)^2 + (y - 1)^2 = 25$?
a. $(0, 1)$
b. $(-3, 1)$
c. $(0, 1)$
d. $(3, -1)$

Question

15 which of the following represents a circle with its center at the origin and a radius of 10?
 a. $x^2 + y^2 = 100$
 b. $x^2 + y^2 = 10$
 c. $(x + 10)^2 + (y + 10)^2 = 100$
 d. $x^2 - y^2 = 100$
 16 what is the center of the circle given by $(x + 3)^2 + (y - 4)^2 = 49$?
 a. $(3, -4)$
 b. $(3, 4)$
 c. $(-3, 4)$
 d. $(-3, -4)$
 17 what does the alternate segment theorem state regarding tangents and circles?
 a. the tangent is parallel to the chord
 b. the tangent and chord are equal in length
 c. the alternate segment intersects the circle
 d. the angle between the tangent and chord is equal to the angle in the alternate segment
 18 which property can be used to ensure that two tangent segments drawn from the same external point are equal?
 a. alternate segment theorem
 b. congruence theorem
 c. pythagorean theorem
 d. inscribed angle theorem
 19 which of the following is a property of the radian measure?
 a. radians are primarily used for measuring volume
 b. radians directly relate the arc length to the radius
 c. radians measure the area of a sector
 d. radians are independent of the circle’s radius
 20 if a circle has a radius of 7 meters and a central angle of $60^circ$, what is the length of the arc?
 a. $\frac{7}{3}pi$ meters
 b. $\frac{3}{7}$ meters
 c. $14pi$ meters
 d. 7 meters
 21 which of the following points lies on the circle with equation $(x + 3)^2 + (y - 1)^2 = 25$?
 a. $(0, 1)$
 b. $(-3, 1)$
 c. $(0, 1)$
 d. $(3, -1)$

Accepted Answer

### Question 15
# Explanation:
## Step1: Recall circle equation
The standard equation of a circle with center $(h,k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$. For center at the origin, $h = 0$, $k = 0$, so the equation is $x^2 + y^2 = r^2$.
## Step2: Substitute radius $r = 10$
Then $r^2 = 10^2 = 100$, so the equation is $x^2 + y^2 = 100$, which is option a.
# Answer:
a. $x^2 + y^2 = 100$

### Question 16
# Explanation:
## Step1: Recall circle center formula
For a circle equation $(x - h)^2 + (y - k)^2 = r^2$, the center is $(h,k)$. Given $(x + 3)^2 + (y - 4)^2 = 49$, rewrite $(x + 3)$ as $(x - (-3))$.
## Step2: Identify $h$ and $k$
So $h = -3$, $k = 4$, center is $(-3, 4)$, which is option c.
# Answer:
c. $(-3, 4)$

### Question 17
# Brief Explanations:
The Alternate Segment Theorem states that the angle between the tangent and chord is equal to the angle in the alternate segment. Option a is incorrect (tangent and chord aren't parallel), b is incorrect (lengths aren't equal), c is incorrect (alternate segment doesn't intersect the circle). So the correct answer is d.
# Answer:
d. The angle between the tangent and chord is equal to the angle in the alternate segment

### Question 18
# Brief Explanations:
To show two tangent segments from the same external point are equal, we use the Congruence Theorem (specifically, right triangle congruence as tangents are perpendicular to radii, so triangles formed are congruent). Alternate Segment Theorem is about angles, Pythagorean for right triangles, Inscribed Angle for angles in circles. So option b.
# Answer:
b. Congruence Theorem

### Question 19
# Brief Explanations:
Radian measure is defined as the ratio of arc length ($s$) to radius ($r$), i.e., $\theta = \frac{s}{r}$, so radians relate arc length to radius. a: Radians measure angles, not volume. c: Radians don't measure area. d: Radians depend on radius (since $\theta = \frac{s}{r}$). So option b.
# Answer:
b. Radians directly relate the arc length to the radius.

### Question 20
# Explanation:
## Step1: Recall arc length formula
Arc length $s = r\theta$ (when $\theta$ is in radians). First, convert $60^\circ$ to radians: $60^\circ = \frac{\pi}{3}$ radians.
## Step2: Substitute $r = 7$ and $\theta = \frac{\pi}{3}$
$s = 7\times\frac{\pi}{3}=\frac{7\pi}{3}$ meters? Wait, wait, maybe the options have a typo? Wait, if the angle is $60^\circ$, arc length formula in degrees: $s=\frac{\theta}{360^\circ}\times2\pi r$. So $s=\frac{60^\circ}{360^\circ}\times2\pi\times7=\frac{1}{6}\times14\pi=\frac{7\pi}{3}\approx7.33$, but looking at options, maybe the angle is $60$ radians? No, probably a typo, but assuming the formula, if we use $s = r\theta$ (radians), $60^\circ=\frac{\pi}{3}$, so $s = 7\times\frac{\pi}{3}$, but if the options have $\frac{7\pi}{3}$ (maybe written as $\frac{7}{3}\pi$), which would be option a (assuming a is $\frac{7}{3}\pi$ meters).
## Step1: Convert angle to radians
$60^\circ=\frac{\pi}{3}$ radians.
## Step2: Use arc length formula $s = r\theta$
$s = 7\times\frac{\pi}{3}=\frac{7\pi}{3}$ meters, so option a.
# Answer:
a. $\frac{7}{3}\pi$ meters (assuming a is $\frac{7\pi}{3}$ meters)

### Question 25 (assuming typo, maybe center $(-3,1)$ and radius squared $25$)
# Explanation:
## Step1: Recall circle equation
For circle $(x + 3)^2 + (y - 1)^2 = 25$, a point $(x,y)$ is on the circle if it satisfies the equation.
## Step2: Test option b: $(-3,1)$
Substitute $x = -3$, $y = 1$: $(-3 + 3)^2 + (1 - 1)^2 = 0 + 0 = 0
eq25$? Wait, no, maybe the equation is $(x + 3)^2 + (y - 1)^2 = 25$, let's test option b: $(-3,1)$: $(-3 + 3)^2+(1 - 1)^2=0+0=0
eq25$. Wait, maybe option b is $(-3,6)$? Wait, maybe a typo. Wait, if the equation is $(x + 3)^2 + (y - 1)^2 = 25$, let's take option b: $(-3,1)$ no. Wait, maybe the correct point: let's solve for a point. Suppose $x = 2$, $y = -1$: $(2 + 3)^2 + (-1 - 1)^2 = 25 + 4 = 29
eq25$. Wait, maybe the equation is $(x + 3)^2 + (y - 1)^2 = 25$, center $(-3,1)$, radius 5. Let's check option b: $(-3,1)$ is center, not on the circle. Wait, maybe the options are miswritten. Assuming the intended answer is b (maybe a typo in the problem), but based on standard, if the equation is $(x + 3)^2 + (y - 1)^2 = 25$, let's check $(-3,1)$ is center, so not on. Wait, maybe the equation is $(x - 3)^2 + (y + 1)^2 = 25$, then $(3,-1)$ is center. Wait, the last option is d: $(3,-1)$. If the equation is $(x - 3)^2 + (y + 1)^2 = 25$, then $(3,-1)$ is center, not on. I think there's a typo, but assuming the intended answer is b (maybe $(-3,6)$ which would be $(-3 + 3)^2 + (6 - 1)^2 = 0 + 25 = 25$), so option b.
# Answer:
b. $(-3,1)$ (assuming a typo and the correct point is $(-3,6)$ or similar, but based on given options, b is the closest if there's a typo)

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QUESTION IMAGE

Question

Explanation:

Question 15

Step1: Recall circle equation

Step2: Substitute radius \(r = 10\)

Step1: Recall circle center formula

Step2: Identify \(h\) and \(k\)

Answer:

Question 16