Question

1. which of the following is a practical application of the distance - formula in astronomy?

a. calculating the distances between celestial bodies
b. determining the color of stars
c. measuring the cost of the sun
d. counting the number of planets in a galaxy

1. a quadrilateral has vertices as a(1,2), b(4,5), c(7,8), and d(4, - 1). which sides are perpendicular?

a. none of the sides are perpendicular
b. ab and bc
c. bc and cd
d. ab and ad

1. the distance between (10,15) and (2,9) is

a. 9 units
b. 11 units
c. 10 units
d. 12 units

1. if two parallel lines r and s are cut by a transversal t, and ∠1 = 140°, what is the measure of the corresponding angle ∠2?

a. 150°
b. 120°
c. 130°
d. 140°

1. what is the key characteristic of a regular polygon inscribed in a circle?

a. all interior angles are acute.
b. all sides are different lengths.
c. all vertices lie on the circle and all sides are equal.
d. all vertices lie inside the circle.

1. in a parallelogram, the measure of two consecutive angles are 3x + 10° and 2x+5°. what is the value of x?

a. x = 35
b. x = 33
c. x = 30
d. x = 32

1. two angles form a linear - pair. if one angle measures 3x - 5 and the other measures 2x + 35, what is the value of x?

a. 30
b. 26
c. 28
d. 15

Explanation:

19.

The distance formula in astronomy is used for calculating distances between celestial bodies. Measuring the size of stars, counting planets, or determining the color of the sun are not applications of the distance formula.

Step1: Find slopes of sides

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For side $AB$ with $A(1,2)$ and $B(4,5)$, $m_{AB}=\frac{5 - 2}{4 - 1}=1$. For side $BC$ with $B(4,5)$ and $C(7,8)$, $m_{BC}=\frac{8 - 5}{7 - 4}=1$. For side $CD$ with $C(7,8)$ and $D(4,-1)$, $m_{CD}=\frac{-1 - 8}{4 - 7}=3$. For side $AD$ with $A(1,2)$ and $D(4,-1)$, $m_{AD}=\frac{-1 - 2}{4 - 1}=-1$.

Step2: Check perpendicularity

Two lines are perpendicular if the product of their slopes is - 1. $m_{AB}\times m_{AD}=1\times(-1)=-1$.

Step1: Apply distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $x_1 = 10,y_1 = 15,x_2 = 2,y_2 = 9$. So $d=\sqrt{(2 - 10)^2+(9 - 15)^2}=\sqrt{(-8)^2+(-6)^2}=\sqrt{64 + 36}=\sqrt{100}=10$.

Answer:

a. Calculating the distances between celestial bodies

20.