Question

1. find the distance between the points (3, -1) and (-4, 1). round to the nearest tenth if necessary. a. 5.4 b. 8.1 c. 7.3 d. 3.5
2. are points c, j, and b collinear or noncollinear?
3. identify the hypothesis and conclusion of this conditional statement: if tomorrow is monday, then yesterday was saturday. a. hypothesis: tomorrow is monday. conclusion: yesterday was not saturday. b. hypothesis: tomorrow is monday. conclusion: yesterday was saturday. c. hypothesis: yesterday was saturday. conclusion: tomorrow is monday. d. hypothesis: yesterday was saturday. conclusion: tomorrow is monday.
4. what is the contrapositive of the following conditional? if a point is in the fourth quadrant, then its coordinates are negative. a. if the coordinates of a point are negative, then the point is in the fourth quadrant. b. if a point is not in the fourth quadrant, then the coordinates of the point are not negative. c. if the coordinates of a point are not negative, then the point is not in the fourth quadrant. d. if a point is in the fourth quadrant, then its coordinates are negative.
5. what is the equation of the line that that is parallel to y = -2x - 5 and passes through the point (3, -5)? a. y = -2x - 4 b. y = -2x + 3 c. y = -2x + 1 d. y = -2x + 7
6. tell whether the lines for each pair of equations are parallel, perpendicular, or neither. y = \frac{5}{3}x + 3 20x + 12y = 12 a. parallel b. perpendicular c. neither

Explanation:

Step1: Recall 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, $(x_1,y_1)=(3,-1)$ and $(x_2,y_2)=(-4,1)$.

Step2: Substitute values

$d=\sqrt{(-4 - 3)^2+(1-(-1))^2}=\sqrt{(-7)^2+(2)^2}=\sqrt{49 + 4}=\sqrt{53}\approx7.3$.

Step3: Analyze collinearity for question 15

Points $C$, $J$, and $B$ are non - collinear as they do not lie on the same straight line in the given rectangular - box figure.

Step4: Identify hypothesis and conclusion for question 16

In a conditional statement "If $p$, then $q$", $p$ is the hypothesis and $q$ is the conclusion. For "If tomorrow is Monday, then yesterday was Saturday", the hypothesis is "Tomorrow is Monday" and the conclusion is "Yesterday was Saturday".

Step5: Find contrapositive for question 17

The contrapositive of a conditional statement "If $p$, then $q$" is "If not $q$, then not $p$". For "If a point is in the fourth quadrant, then its coordinates are negative", the contrapositive is "If the coordinates of a point are not negative, then the point is not in the fourth quadrant".

Step6: Find equation of parallel line for question 18

Parallel lines have the same slope. The slope of $y=-2x - 5$ is $m=-2$. Using the point - slope form $y - y_1=m(x - x_1)$ with $(x_1,y_1)=(3,-5)$ and $m=-2$, we have $y+5=-2(x - 3)$, which simplifies to $y=-2x+1$.

Step7: Determine line relationship for question 19

Rewrite $20x + 12y=12$ in slope - intercept form $y=mx + b$: $12y=-20x + 12$, so $y=-\frac{5}{3}x + 1$. The slope of $y=\frac{5}{3}x+3$ is $m_1=\frac{5}{3}$ and the slope of $y =-\frac{5}{3}x + 1$ is $m_2=-\frac{5}{3}$. Since $m_1
eq m_2$ and $m_1\times m_2
eq - 1$, the lines are neither parallel nor perpendicular.

Answer:

1. c. 7.3
2. a. noncollinear
3. b. Hypothesis: Tomorrow is Monday. Conclusion: Yesterday was Saturday.
4. c. If the coordinates of a point are not negative, then the point is not in the fourth quadrant.
5. c. $y=-2x + 1$
6. c. neither