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 P(2,1)P(2, -1) and Q(3,1)Q(3, 1), we proceed as follows:


Parametric Equations:

A parametric equation for the line segment can be written as: x(t)=(1t)xP+txQx(t) = (1-t)x_P + t x_Q y(t)=(1t)yP+tyQy(t) = (1-t)y_P + t y_Q where t[0,1]t \in [0, 1], and tt is the parameter that moves from PP to QQ.

Substituting the coordinates of P(2,1)P(2, -1) and Q(3,1)Q(3, 1): x(t)=(1t)(2)+t(3)x(t) = (1-t)(2) + t(3) y(t)=(1t)(1)+t(1)y(t) = (1-t)(-1) + t(1)

Simplify: x(t)=22t+3t=2+tx(t) = 2 - 2t + 3t = 2 + t y(t)=1t+t(1)=1+2ty(t) = -1 - t + t(1) = -1 + 2t

Thus, the parametric equations for the line segment are: x(t)=2+t,y(t)=1+2t,t[0,1]x(t) = 2 + t, \quad y(t) = -1 + 2t, \quad t \in [0, 1]


Additional Points on the Line Segment:

Using these parametric equations, we find specific points by substituting values of tt for each case:


a) The midpoint between PP and QQ (t=0.5t = 0.5):

Substitute t=0.5t = 0.5 into the parametric equations: x(0.5)=2+0.5=2.5x(0.5) = 2 + 0.5 = 2.5 y(0.5)=1+2(0.5)=0y(0.5) = -1 + 2(0.5) = 0

The midpoint is: (2.5,0)(2.5, 0)


b) The point that is one-fourth of the way from PP to QQ (t=0.25t = 0.25):

Substitute t=0.25t = 0.25: x(0.25)=2+0.25=2.25x(0.25) = 2 + 0.25 = 2.25 y(0.25)=1+2(0.25)=0.5y(0.25) = -1 + 2(0.25) = -0.5

The point is: (2.25,0.5)(2.25, -0.5)


c) The point that is three-fourths of the way from PP to QQ (t=0.75t = 0.75):

Substitute t=0.75t = 0.75: x(0.75)=2+0.75=2.75x(0.75) = 2 + 0.75 = 2.75 y(0.75)=1+2(0.75)=0.5y(0.75) = -1 + 2(0.75) = 0.5

The point is: (2.75,0.5)(2.75, 0.5)


Final Results:

  1. Parametric equations: x(t)=2+tx(t) = 2 + t, y(t)=1+2ty(t) = -1 + 2t, t[0,1]t \in [0, 1]
  2. Midpoint: (2.5,0)(2.5, 0)
  3. One-fourth point: (2.25,0.5)(2.25, -0.5)
  4. Three-fourths point: (2.75,0.5)(2.75, 0.5)

Let me know if you want further details or clarifications. Here are 5 related questions to explore:

  1. How would the parametric equations change for a line segment in 3D space?
  2. How do you calculate the length of a line segment using parametric equations?
  3. Can you find the equation of a line perpendicular to PQPQ that passes through the midpoint?
  4. How do parametric equations relate to vector equations of a line?
  5. What if t(,)t \in (-\infty, \infty); 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