8. how many radians are there in a quarter of a circle?
a. $\pi/6$ radians
b. $\pi/3$ radians
c. $\pi/8$ radians
d. $\pi/2$ radians

9. what is the center of a circle represented by $(x - 4)^2 + (y + 7)^2 = 81$?
a. $(-4, -7)$
b. $(4, -7)$
c. $(4, 7)$
d. $(-4, 7)$

10. how do you rewrite the equation $x^2 + y^2 + 6x - 8y - 11 = 0$ in standard form?
a. $(x - 3)^2 + (y + 4)^2 = 36$
b. $(x + 3)^2 + (y - 4)^2 = 36$
c. $(x - 3)^2 + (y - 4)^2 = 36$
d. $(x + 3)^2 + (y + 4)^2 = 36$

11. which of the following describes the circumcenter in a right triangle?
a. it coincides with the centroid
b. it divides the triangle into three smaller triangles of equal area
c. it is the lowest point of the triangle
d. it is located at the midpoint of the hypotenuse

12. what is true about the circumcenter of a triangle?
a. it is the same as the incenter
b. it is equidistant from all vertices
c. it is always inside the triangle
d. it is always on the hypotenuse

13. if a central angle measures $2\pi/5$ radians in a circle with radius 5 cm, what is the arc length?
a. $5\pi$ cm
b. $2\pi$ cm
c. $3\pi$ cm
d. $\pi$ cm

14. which of the following points are used in constructing the incircle?
a. incenter
b. tangent
c. circumcenter
d. centroid

Question

8. how many radians are there in a quarter of a circle?
 a. $\pi/6$ radians
 b. $\pi/3$ radians
 c. $\pi/8$ radians
 d. $\pi/2$ radians

9. what is the center of a circle represented by $(x - 4)^2 + (y + 7)^2 = 81$?
 a. $(-4, -7)$
 b. $(4, -7)$
 c. $(4, 7)$
 d. $(-4, 7)$

10. how do you rewrite the equation $x^2 + y^2 + 6x - 8y - 11 = 0$ in standard form?
 a. $(x - 3)^2 + (y + 4)^2 = 36$
 b. $(x + 3)^2 + (y - 4)^2 = 36$
 c. $(x - 3)^2 + (y - 4)^2 = 36$
 d. $(x + 3)^2 + (y + 4)^2 = 36$

11. which of the following describes the circumcenter in a right triangle?
 a. it coincides with the centroid
 b. it divides the triangle into three smaller triangles of equal area
 c. it is the lowest point of the triangle
 d. it is located at the midpoint of the hypotenuse

12. what is true about the circumcenter of a triangle?
 a. it is the same as the incenter
 b. it is equidistant from all vertices
 c. it is always inside the triangle
 d. it is always on the hypotenuse

13. if a central angle measures $2\pi/5$ radians in a circle with radius 5 cm, what is the arc length?
 a. $5\pi$ cm
 b. $2\pi$ cm
 c. $3\pi$ cm
 d. $\pi$ cm

14. which of the following points are used in constructing the incircle?
 a. incenter
 b. tangent
 c. circumcenter
 d. centroid

Accepted Answer

### Question 8
# Explanation:
## Step1: Recall the total radians in a circle
A full circle is $2\pi$ radians.
## Step2: Calculate a quarter of a circle
A quarter means $\frac{1}{4}$ of the total. So, $\frac{1}{4} \times 2\pi=\frac{\pi}{2}$ radians.
# Answer:
d. $\pi/2$ radians

### Question 9
# Explanation:
## Step1: Recall the standard circle equation
The standard form of a circle's equation is $(x - h)^2+(y - k)^2 = r^2$, where $(h,k)$ is the center.
## Step2: Identify $h$ and $k$ from the given equation
For the equation $(x - 4)^2+(y + 7)^2 = 81$, we can rewrite $y + 7$ as $y-(-7)$. So, $h = 4$ and $k=-7$.
# Answer:
b. $(4,-7)$

### Question 10
# Explanation:
## Step1: Start with the given equation
The equation is $x^{2}+y^{2}+6x - 8y-11 = 0$.
## Step2: Group $x$ and $y$ terms
Group the $x$ terms and $y$ terms: $(x^{2}+6x)+(y^{2}-8y)=11$.
## Step3: Complete the square for $x$ terms
For $x^{2}+6x$, take half of 6 (which is 3) and square it: $3^{2}=9$. Add 9 to both sides.
## Step4: Complete the square for $y$ terms
For $y^{2}-8y$, take half of - 8 (which is - 4) and square it: $(-4)^{2}=16$. Add 16 to both sides.
## Step5: Rewrite the equation
We get $(x^{2}+6x + 9)+(y^{2}-8y + 16)=11 + 9+16$. Which simplifies to $(x + 3)^{2}+(y - 4)^{2}=36$ (since $x^{2}+6x + 9=(x + 3)^{2}$ and $y^{2}-8y + 16=(y - 4)^{2}$ and $11 + 9+16 = 36$).
# Answer:
b. $(x + 3)^{2}+(y - 4)^{2}=36$

### Question 11
# Explanation:
## Step1: Recall the definition of circumcenter
The circumcenter is the intersection of the perpendicular bisectors of the triangle's sides.
## Step2: Recall properties of right - triangle circumcenter
In a right triangle, the circumcenter is located at the mid - point of the hypotenuse. Let's analyze the options:
- Option a: The centroid is the intersection of medians, not the same as circumcenter in general (and not in a right triangle in this specific way).
- Option b: The centroid divides the triangle into triangles of equal area, not the circumcenter.
- Option c: The circumcenter of a right triangle is not the lowest point (unless the triangle is oriented in a very specific way, which is not a general property).
- Option d: This is the correct property of the circumcenter of a right triangle.
# Answer:
d. It is located at the midpoint of the hypotenuse

### Question 12
# Explanation:
## Step1: Recall the definition of circumcenter
The circumcenter is the point equidistant from all the vertices of the triangle (it is the center of the circumscribed circle around the triangle).
## Step2: Analyze the options
- Option a: The incenter is equidistant from the sides, not the same as circumcenter.
- Option b: This is the correct property of the circumcenter.
- Option c: The circumcenter can be outside the triangle (for obtuse triangles), so it is not always inside.
- Option d: The circumcenter is on the hypotenuse only for right triangles, not for all triangles.
# Answer:
b. It is equidistant from all vertices

### Question 13
# Explanation:
## Step1: Recall the arc - length formula
The formula for arc length $s$ is $s=r\theta$, where $r$ is the radius and $\theta$ is the central angle in radians.
## Step2: Substitute the given values
Given $r = 5$ cm and $\theta=\frac{2\pi}{5}$ radians. Then $s=5\times\frac{2\pi}{5}=2\pi$ cm.
# Answer:
b. $2\pi$ cm

### Question 14
# Explanation:
## Step1: Recall the definitions of the points
- Incenter: The center of the inscribed circle, equidistant from the sides.
- Tangent: A line that touches the circle at one point, not a point used to construct the circle.
- Circumcenter: The center of the circumscribed circle (the circle passing through all vertices of the triangle), so it is used to construct the circum - circle.
- Centroid: The intersection of medians, not related to circle construction.
## Step2: Identify the correct point
The circumcenter is used to construct the circum - circle (the circle around the triangle).
# Answer:
c. Circumcenter

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 8

Step1: Recall the total radians in a circle

Step2: Calculate a quarter of a circle

Step1: Recall the standard circle equation

Step2: Identify \(h\) and \(k\) from the given equation

Step1: Start with the given equation

Step2: Group \(x\) and \(y\) terms

Step3: Complete the square for \(x\) terms

Step4: Complete the square for \(y\) terms

Step5: Rewrite the equation

Answer:

Question 9