Help?
1. In your own words, explain why we use the formula AB = |a - b| to find the distance of a segment. Include why you need to subtract and why you need to take the absolute value.
2. Compare and Contrast Describe the difference between saying that two segments are congruent and saying that two
segments have equal length. When would you use each phrase?
1. To understand why we use the formula AB = |a - b| to find the distance of a segment, let's break it down step by step:
- The formula starts with subtracting the coordinates. The coordinates ‘a’ and ‘b’ represent the endpoints of the segment we are interested in finding the distance of.
- By subtracting the coordinates, we are essentially calculating the difference between the two points. This gives us the length of the segment.
- However, this difference can be either positive or negative, depending on the orientation of the endpoints. For example, if point ‘a’ is at (3, 2) and point ‘b’ is at (6, 5), the length is positive. But if the points were reversed, with ‘a’ at (6, 5) and ‘b’ at (3, 2), the length would be negative.
- To overcome this issue, we need to take the absolute value of the difference. The absolute value eliminates the negativeness, ensuring that our distance value is always positive.
In summary, the formula AB = |a - b| is used because it subtracts the coordinates to find the difference in length between two points, and then takes the absolute value to ensure a positive distance regardless of the order of the endpoints.
2. While the phrases "two segments are congruent" and "two segments have equal length" might sound similar, there is a difference in their usage:
- "Two segments are congruent" means that the segments have the same length and the same orientation. In other words, if you were to physically place one segment on top of the other, they would perfectly align, indicating that they are identical in length and shape.
- On the other hand, "two segments have equal length" simply means that the lengths of the segments are the same, regardless of their orientation or shape. They may not be identical in shape, but they share the same measurement.
In the context of usage, you would use "two segments are congruent" when you want to state that two segments are visually identical in length and shape. This is often used in geometric proofs or when comparing figures.
Meanwhile, you would use "two segments have equal length" when you want to emphasize that the lengths of two segments are equivalent, without concerning their specific shape or orientation. This can be used in various mathematical contexts, such as equations or when comparing measurements.