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Parallel and Perpendicular Lines: Mastery Test

Select the correct answer.

Two objects are moving along separate linear paths where each path is described by position, [tex]$d$[/tex], and time, [tex]$t$[/tex]. The variable [tex]$d$[/tex] is measured in meters and the variable [tex]$t$[/tex] is measured in seconds. The equation describing the position of the first object with respect to time is [tex]$d = 2.5t$[/tex]. The graph of the position of the second object is a parallel line passing through [tex]$(t=0, d=1)$[/tex]. What is the equation of the second line?

A. [tex]$d=2.5 t+1$[/tex]
B. [tex]$d=t+25$[/tex]
C. [tex]$d=-0.4 t+1$[/tex]
D. [tex]$d=2.5 t+3.2$[/tex]

Sagot :

GinnyAnswer
To determine the equation of the second object's path that is parallel to the first object and passes through the point [tex]\((t=0, d=1)\)[/tex], we need to understand a few key concepts about parallel lines in coordinate geometry.

### Key Concepts:

1. Parallel Lines: Two lines are parallel if and only if they have the same slope.
2. Slope-Intercept Form: The general form of a linear equation is [tex]\(d = mt + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.

### Given:

- The second object’s line must pass through the point [tex]\((t=0, d=1)\)[/tex].
- The lines are described as parallel.

### Steps to Solve:

1. Identify the Y-Intercept:
Since the second object's line passes through the point [tex]\((t=0, d=1)\)[/tex], the y-intercept [tex]\(b\)[/tex] is 1. Therefore, the correct equation will have a y-intercept of 1.

2. Check Parallel Lines:
Since the lines are parallel, they will have the same slope. We need to identify the option which has both a y-intercept of 1 and a slope that matches the slope of the first line.

### Evaluating Each Option:

- Option A: [tex]\(d = 2.5t + 1\)[/tex]
- Slope [tex]\(= 2.5\)[/tex]
- Y-intercept [tex]\(= 1\)[/tex]
- This line is parallel and passes through [tex]\((0,1)\)[/tex].

- Option B: [tex]\(d = t + 25\)[/tex]
- Slope [tex]\(= 1\)[/tex]
- Y-intercept [tex]\(= 25\)[/tex]
- This line has the wrong y-intercept and a different slope.

- Option C: [tex]\(d = -0.4t + 1\)[/tex]
- Slope [tex]\(= -0.4\)[/tex]
- Y-intercept [tex]\(= 1\)[/tex]
- Although it passes through [tex]\((0, 1)\)[/tex], it is not parallel due to the different slope.

- Option D: [tex]\(d = 2.5t + 3.2\)[/tex]
- Slope [tex]\(= 2.5\)[/tex]
- Y-intercept [tex]\(= 3.2\)[/tex]
- This line has the correct slope but does not pass through [tex]\((0, 1)\)[/tex].

### Conclusion:

The only equation that satisfies both conditions — having the same slope and the correct y-intercept — is Option A: [tex]\(d = 2.5t + 1\)[/tex].

Thus, the correct answer is:

A. [tex]\(d=2.5 t+1\)[/tex]
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