High School

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------------------------------------------------ The following statements describe a set of railroad tracks. Based only on the statement, make a conclusion about the rails or the railroad ties. Explain.

a. The railroad ties are each perpendicular to one rail.

b. The rails are parallel. One railroad tie is perpendicular to one rail.

Answer :

Answer:

Part a) The railroad ties are parallel


Part b) One railroad tie is perpendicular to both rails

Step-by-step explanation:

Part a) The railroad ties are each perpendicular to one rail

we know that

if one line is perpendicular to several other lines, then the other lines are parallel

Conclusion:

If the railroad ties are each perpendicular to one rail then the railroad ties are parallel

That means----> The slopes of the railroad ties are equal and they are opposite reciprocal to the slope of the rail

Part b) The rails are parallel. One railroad tie is perpendicular to one rail

we know that

if two lines are parallel and another line is perpendicular to one of them, then this line is perpendicular to both lines

Conclusion:

If One railroad tie is perpendicular to one rail, then one railroad tie is perpendicular to both rails

That means ---> The slopes of the railroad ties are opposite reciprocal to the slope of the rail

Final answer:

The railroad ties are perpendicular to both rails, and the rails are parallel to each other, reflecting principles of linear perspective in geometry.

Explanation:

The student's questions relate to geometry, specifically the properties of lines and angles about railroad tracks and railroad ties. When analyzing these geometric structures, we draw on concepts such as parallel lines, and perpendicular lines, and the characteristics of their interaction.

Analysis of Statements

  • Statement a: If railroad ties are each perpendicular to one rail, and knowing that both rails in a railroad must be parallel, we can conclude that the ties are also perpendicular to the other rail. This is because perpendicular lines or segments are by definition at 90° to each other, and if one tie is perpendicular to one parallel rail it must be perpendicular to the other rail that is parallel to the first.
  • Statement b: If one railroad tie is perpendicular to one rail and the rails are parallel, then every other railroad tie that is also perpendicular to one rail will be perpendicular to both rails, maintaining consistency in the geometry of the railroad structure.

Therefore, we can conclude that the railroad ties are perpendicular to both rails and that the rails are parallel to each other. This aligns with linear perspective and the visual effect of perpendicular lines (orthogonal) converging at a vanishing point as they recede into the distance.

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