Coordinate Geometry: Distance, Midpoint, Collinearity, and Parallelograms

Coordinate Geometry

Distance Between Two Points

The distance formula is used to find the length between two points in the coordinate plane. Given points $P(x_1, y_1)$ and $Q(x_2, y_2)$, the distance $d$ between them is:

• Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Example: For $P(1,5)$ and $Q(1,10)$: $d = \sqrt{(1-1)^2 + (10-5)^2} = \sqrt{0 + 25} = 5$

• Application: This formula is fundamental for calculating lengths, perimeters, and verifying geometric properties in the plane.

Midpoint of a Line Segment

The midpoint formula finds the point exactly halfway between two given points $P(x_1, y_1)$ and $Q(x_2, y_2)$:

• Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Example: For $P(1,5)$ and $Q(1,10)$: $M = \left( \frac{1+1}{2}, \frac{5+10}{2} \right) = (1, 7.5)$

• Application: The midpoint is useful for bisecting segments and constructing geometric figures.

Using Radicals in Answers

Sometimes, the coordinates or distances involve square roots (radicals). Answers should be left in exact form unless specified otherwise.

• Example: For $P(\sqrt{3}, 5)$ and $Q(\sqrt{3}, 10)$: $d = \sqrt{(\sqrt{3} - \sqrt{3})^2 + (10-5)^2} = \sqrt{0 + 25} = 5$ $M = \left( \frac{\sqrt{3} + \sqrt{3}}{2}, \frac{5+10}{2} \right) = (\sqrt{3}, 7.5)$

Collinearity of Points

Three points are collinear if they lie on the same straight line. This can be checked by comparing the sum of the distances between two pairs to the distance between the outer points.

• Criterion: Points $P$, $Q$, $R$ are collinear if $d(P,Q) + d(Q,R) = d(P,R)$.

• Example: For $P=(0,0)$, $Q=(4,7)$, $R=(-4,-7)$: Calculate $d(P,Q)$, $d(Q,R)$, and $d(P,R)$ and check if the sum equals the third.

• Application: Collinearity is important for geometric proofs and constructions.

Equidistant Points on the y-axis

To find a point on the y-axis that is equidistant from two given points, set up the distance formula from the unknown point $(0, y)$ to each given point and solve for $y$.

• Example: Find $y$ such that $(0, y)$ is equidistant from $(-2,0)$ and $(1,3)$: Set $\sqrt{(0+2)^2 + (y-0)^2} = \sqrt{(0-1)^2 + (y-3)^2}$ and solve for $y$.

• Result: $y = 1$

Estimating Values Using Linear Trends

Given data for two years, you can estimate values for other years by assuming a linear trend.

• Example: Americans spent $80$ billion in 1998 and $146$ billion in 2000. Estimate the amount spent in 1998 if the trend is linear.

• Calculation: Find the rate of change and apply it to the desired year.

• Result: $117$ billion in 1998.

• Application: Useful for predictions and interpolations in statistics and economics.

Parallelograms in the Coordinate Plane

A parallelogram is a quadrilateral with opposite sides parallel. In the coordinate plane, you can use properties of diagonals and midpoints to find missing vertices.

• Diagonals Bisect Each Other: The diagonals of a parallelogram bisect each other, meaning their midpoints are the same.

• Finding a Fourth Vertex: Given three vertices $A$, $B$, $C$, you can find $D$ so that $ABCD$ forms a parallelogram by using midpoint or vector methods.

• Example Table:

Given Vertices	Diagonal Used	Coordinates of D
A(2,1), B(7,8), C(2,10)	AC	(7,-1)
A(2,1), B(7,8), C(2,10)	BC	(-3,3)
A(2,1), B(7,8), C(2,10)	AB	(-3,3)
A(3,5), B(7,8), C(1,3)	Parallelogram vertex x	11

• Application: This method is essential for constructing parallelograms and solving geometric problems in analytic geometry.

Summary Table: Key Formulas

Concept	Formula	Example
Distance	$\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$	$P(1,5), Q(1,10): 5$
Midpoint	$\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)$	$P(1,5), Q(1,10): (1,7.5)$
Collinearity	$d(P,Q) + d(Q,R) = d(P,R)$	$P=(0,0), Q=(4,7), R=(-4,-7)$
Parallelogram Diagonals	Diagonals bisect each other	Find $D$ given $A$, $B$, $C$

Additional info: The notes above expand on the brief points in the original material, providing full definitions, formulas, and context for Precalculus students studying coordinate geometry.