Question

1. which is not a point of concurrency in a triangle?

a midpoint of a side
b. circumcenter
c. incenter
d. centroid

1. the circumcircle of a triangle passes through all

a. vertices of the triangle
b. perpendicular bisectors
c. midpoints of the sides
d. angle bisectors

1. if the central angle of a sector is \\( \frac{9}{2} \\) radians and the radius is 8 meters, what is the area of the sector?

a. 36 square meters
b. 144 square meters
c. 324 square meters
d. 648 square meters

1. which of the following formulas is used to calculate the length of an arc in radians?

a. \\( l=\theta r \\)
b. \\( l=2 e \theta \\)
c. \\( l=\theta r^{2} \\)
d. \\( l=\frac{1}{2} \theta r^{2} \\)

1. what is the relationship between the radius and the tangent line at the point of tangency?

a. they are equal in length
b. they are collinear
c. they are parallel
d. they are perpendicular

1. which statement is false about a circles tangent line?

a. it forms a perpendicular angle with the radius at the point of tangency
b. it intersects the circle at exactly one point
c. it never passes through the circles center
d. it forms a 45 - degree angle with the radius at the point of tangency

1. which statement is true about the circumcircle of a triangle?

a. its center is the centroid
b. it is tangent to all three sides
c. it is centered inside the triangle
d. it passes through all three vertices of a triangle

Explanation:

1. A midpoint of a side is just a point on a side, not where three lines concur. Circumcenter, incenter, and centroid are concurrency points of perpendicular bisectors, angle bisectors, and medians respectively.
2. The circumcircle passes through all three vertices of the triangle by definition.
3. Area of sector = $\frac{1}{2}r^2\theta$. Here, $r=8$, $\theta=\frac{5}{2}$, so area = $\frac{1}{2}(8)^2(\frac{5}{2}) = 80$.
4. Arc length formula in radians is $L = r\theta$.
5. Radius and tangent at the point of contact are perpendicular.
6. A tangent does not form a 45-degree angle with the radius (it forms 90 degrees), so this is false.

Answer:

1. a. Midpoint of a side
2. a. Vertices of the triangle
3. a. 80 square meters
4. a. $L = r\theta$
5. d. They are perpendicular
6. d. It forms a 45-degrees angle with the radius at the point of tangency