1. the point where the perpendicular bisectors of a triangle intersect is called the
a. incenter
b. orthocenter
c. circumcenter
d. centroid

2. how many radians are there in a complete circle?
a. $\pi/2$ radians
b. $3\pi$ radians
c. $\pi$ radians
d. $2\pi$ radians

3. which property is true about the incenter of a triangle?
a. it is equidistant from all vertices
b. it is always on the triangle’s altitude
c. it is equidistant from all sides
d. it divides the medians into a 2:1 ratio

4. in an acute triangle, where is the circumcenter located?
a. at the centroid
b. on the hypotenuse
c. inside the triangle
d. outside the triangle

5. which of the following formulae is used to calculate the length of an arc in radians?
a. $l = \pi r^2$
b. $l = \frac{1}{2}ht^2$
c. $l = 2\pi\theta$
d. $l = r\theta$

6. what is the first step to construct the centroid of a triangle?
a. construct the perpendicular bisectors of each side
b. construct the angle bisectors of each angle
c. draw a circle in the triangle
d. identify the midpoints of each side

7. what is the center of the circle with the equation $(x + 8)^2 + (y - 6)^2 = 36$?
a. $(8, 6)$
b. $(-8, -6)$
c. $(8, -6)$
d. $(-8, 6)$

Question

1. the point where the perpendicular bisectors of a triangle intersect is called the
a. incenter
b. orthocenter
c. circumcenter
d. centroid

2. how many radians are there in a complete circle?
a. $\pi/2$ radians
b. $3\pi$ radians
c. $\pi$ radians
d. $2\pi$ radians

3. which property is true about the incenter of a triangle?
a. it is equidistant from all vertices
b. it is always on the triangle’s altitude
c. it is equidistant from all sides
d. it divides the medians into a 2:1 ratio

4. in an acute triangle, where is the circumcenter located?
a. at the centroid
b. on the hypotenuse
c. inside the triangle
d. outside the triangle

5. which of the following formulae is used to calculate the length of an arc in radians?
a. $l = \pi r^2$
b. $l = \frac{1}{2}ht^2$
c. $l = 2\pi\theta$
d. $l = r\theta$

6. what is the first step to construct the centroid of a triangle?
a. construct the perpendicular bisectors of each side
b. construct the angle bisectors of each angle
c. draw a circle in the triangle
d. identify the midpoints of each side

7. what is the center of the circle with the equation $(x + 8)^2 + (y - 6)^2 = 36$?
a. $(8, 6)$
b. $(-8, -6)$
c. $(8, -6)$
d. $(-8, 6)$

Accepted Answer

### Question 1
# Brief Explanations:
The point where perpendicular bisectors of a triangle intersect is the circumcenter. Incenter is where angle bisectors meet, centroid is where medians meet, and orthocenter is where altitudes meet. So the correct option is related to circumcenter.
# Answer:
b. Circumcenter (assuming option b is labeled as Circumcenter, as per the context of the question about perpendicular bisectors intersection)

### Question 2
# Brief Explanations:
A complete circle (360 degrees) is equal to $2\pi$ radians. $\pi/2$ is 90 degrees, $3\pi$ is 540 degrees, $\pi$ is 180 degrees. So the correct answer is $2\pi$ radians.
# Answer:
d. $2\pi$ radians

### Question 3
# Brief Explanations:
The incenter of a triangle is equidistant from all sides (it's the center of the incircle, tangent to all sides). Equidistant from vertices is circumcenter, on altitude is not a property of incenter, and dividing medians in 2:1 is centroid. So the correct property is being equidistant from all sides.
# Answer:
c. It is equidistant from all sides

### Question 4
# Brief Explanations:
In an acute triangle, the circumcenter (intersection of perpendicular bisectors) lies inside the triangle. In a right triangle, it's on the hypotenuse, and in an obtuse triangle, it's outside. Centroid is always inside but the question is about circumcenter. So for acute, it's inside.
# Answer:
c. Inside the triangle

### Question 5
# Brief Explanations:
The formula for arc length in radians is $L = r\theta$ (where $r$ is radius, $\theta$ is central angle in radians). $E = mv^2$ is kinetic energy, $L=\frac{1}{2}gt^2$ is for free - fall distance, $L = 2\pi\theta$ is incorrect. So the correct formula is $L=r\theta$.
# Answer:
d. $L = r\theta$

### Question 6
# Brief Explanations:
To construct the centroid of a triangle, we first need to find the midpoints of each side (since centroid is the intersection of medians, and a median connects a vertex to the midpoint of the opposite side). Constructing perpendicular bisectors is for circumcenter, angle bisectors for incenter, and drawing a circle is not the first step for centroid. So the first step is to identify the midpoints of each side.
# Answer:
d. Identify the midpoints of each side

### Question 7
# Brief Explanations:
The standard form of a circle's equation is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center. For the equation $(x + 8)^2+(y - 6)^2 = 36$, we can rewrite $x + 8$ as $x-(-8)$. So $h=-8$ and $k = 6$, so the center is $(-8,6)$.
# Answer:
d. $(-8,6)$

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