Question

1. what is the area of a sector with a radius of 10 meters and a central angle of 45°?

a. $\frac{25pi}{2}$ square meters
b. $25pi$ square meters
c. $\frac{50pi}{2}$ square meters
d. $50pi$ square meters

1. if two triangles have side - ratios $\frac{1}{3},\frac{1}{3},\frac{1}{3}$, what is their scale factor?

a. $\frac{1}{3}$
b. 1
c. 3
d. 2

1. given a parallelogram with vertices at $a(1,1),b(5,1),c(6,4)$, and $d(2,4)$, which calculation proves that the diagonals bisect each other?

a. slopes of the diagonals are equal
b. slopes of opposite sides are equal
c. diagonal lengths are equal
d. mid - points of diagonals are equal

Explanation:

Step1: Recall sector - area formula

The formula for the area of a sector of a circle is $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$, where $\theta$ is the central - angle measure in degrees and $r$ is the radius of the circle.

Step2: Substitute given values

Given $r = 10$ meters and $\theta=45^{\circ}$, we substitute into the formula: $A=\frac{45^{\circ}}{360^{\circ}}\times\pi\times(10)^{2}$.

Step3: Simplify the expression

$\frac{45}{360}=\frac{1}{8}$, and $(10)^{2}=100$. So $A=\frac{1}{8}\times\pi\times100=\frac{25\pi}{2}$ square meters.

Step4: Recall scale - factor concept

If the side - ratios of two similar triangles are $\frac{1}{3},\frac{1}{3},\frac{1}{3}$, the scale factor is the ratio of the corresponding side lengths of the two similar triangles. The scale factor $k$ is such that if the side lengths of one triangle are $a,b,c$ and the side lengths of the other are $a',b',c'$, then $\frac{a}{a'}=\frac{b}{b'}=\frac{c}{c'}=k$. Here, $k = \frac{1}{3}$.

Step5: Recall the property of bisecting diagonals in a parallelogram

The mid - points of the diagonals of a parallelogram are equal if and only if the diagonals bisect each other. The slope of a line segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$, and the length of a line segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}$. The mid - point of a line segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

Answer:

1. C. $\frac{25\pi}{2}$ square meters
2. A. $\frac{1}{3}$
3. D. Midpoints of diagonals are equal