Math Problem Statement
Find parametric equations for the line segment joining P(2, −1) and Q(3, 1), and using parameter form of the line segment also find a) the midpoint between P and Q, b) the point that is one-fourth of the way from P to Q, c) the point that is three-fourths of the way from P to Q.
Solution
To find the parametric equations for the line segment joining and , we proceed as follows:
Parametric Equations:
A parametric equation for the line segment can be written as: where , and is the parameter that moves from to .
Substituting the coordinates of and :
Simplify:
Thus, the parametric equations for the line segment are:
Additional Points on the Line Segment:
Using these parametric equations, we find specific points by substituting values of for each case:
a) The midpoint between and ():
Substitute into the parametric equations:
The midpoint is:
b) The point that is one-fourth of the way from to ():
Substitute :
The point is:
c) The point that is three-fourths of the way from to ():
Substitute :
The point is:
Final Results:
- Parametric equations: , ,
- Midpoint:
- One-fourth point:
- Three-fourths point:
Let me know if you want further details or clarifications. Here are 5 related questions to explore:
- How would the parametric equations change for a line segment in 3D space?
- How do you calculate the length of a line segment using parametric equations?
- Can you find the equation of a line perpendicular to that passes through the midpoint?
- How do parametric equations relate to vector equations of a line?
- What if ; how does this affect the parametric form?
Tip: Parametric equations are powerful tools for modeling curves, not just straight lines, and are widely used in computer graphics and physics!
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Parametric Equations
Linear Interpolation
Formulas
Parametric equations: x(t) = (1-t)x_P + t x_Q, y(t) = (1-t)y_P + t y_Q
Midpoint formula: (x_mid, y_mid) = ((x1 + x2)/2, (y1 + y2)/2)
Theorems
Linear Interpolation
Suitable Grade Level
Grades 9-12
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