Math Problem Statement
- For two points I and j on a plane surface, derive the observation for distance between them in plane coordinates. If the coordinates of point I are said to be fixed, what is the implication of this statement and reflect or show it in the observation equation for distance earlier derived. ( Hint: There is no guess attempt. Take distance ( ) i.e. )
Solution
To derive the distance between two points and on a plane surface in plane coordinates, we use the Euclidean distance formula. Let's denote the coordinates of point as and the coordinates of point as .
The distance between points and is given by:
Derivation
-
Identify Coordinates:
- Point :
- Point :
-
Calculate the Differences in Coordinates:
- Difference in x-coordinates:
- Difference in y-coordinates:
-
Apply the Pythagorean Theorem:
- According to the Pythagorean theorem in a right triangle, the square of the hypotenuse (distance ) is the sum of the squares of the other two sides (differences in coordinates).
- Thus,
-
Take the Square Root:
- Substituting and :
Fixed Point
If the coordinates of point are fixed, it means that and are constants. This has the following implications for the observation equation:
-
Fixed Coordinates:
- Since and are fixed, they do not change. Only the coordinates of point ( and ) are variable.
-
Simplified Observation Equation:
- The observation equation remains the same, but with and as constants:
- This implies that for any given point , the distance is a function of the coordinates of relative to the fixed point .
Observation Equation for Fixed Point :
Here, and are fixed values, and is computed based on the coordinates of point .
Would you like further details or have any questions about this derivation?
Here are 5 more questions you might consider asking:
- How do you derive the midpoint between two points in a plane?
- What is the formula for the slope of the line passing through two points?
- How do you find the equation of a line given two points?
- Can you explain how to calculate the area of a triangle given its vertices in plane coordinates?
- What is the distance formula in three-dimensional space?
Tip: Always double-check your coordinate differences before substituting into the distance formula to avoid sign errors.
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Math Problem Analysis
Mathematical Concepts
Euclidean Geometry
Coordinate Geometry
Formulas
Distance formula in Euclidean space
Theorems
Pythagorean theorem
Suitable Grade Level
Grades 9-12
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