08 Analytic Geometry

The coordinate framework

Analytic geometry connects geometric figures with coordinates, equations, and algebra. A geometric condition such as equal length, parallelism, perpendicularity, or collinearity can often be translated into a calculation.

Locating points and reading coordinates

The coordinate plane has a horizontal $x$-axis and a vertical $y$-axis. Their intersection is the origin $(0,0)$. A point $(x,y)$ records horizontal displacement first and vertical displacement second.

The signs of the coordinates identify the quadrants:

• Quadrant I: $x>0$ and $y>0$.

• Quadrant II: $x<0$ and $y>0$.

• Quadrant III: $x<0$ and $y<0$.

• Quadrant IV: $x>0$ and $y<0$.

For example, $(-3,2)$ lies in Quadrant II. A point with $y=0$ is an $x$-intercept, and a point with $x=0$ is a $y$-intercept. Points on an axis are not in any quadrant.

Takeaway: Always interpret an ordered pair as $(x,y)$, and use the signs of its coordinates to locate the point.

Measuring coordinate distance

The distance formula is obtained by applying the Pythagorean theorem to the horizontal and vertical changes between two points. For $A(x_1,y_1)$ and $B(x_2,y_2)$,

For $A(-3,-1)$ and $B(2,3)$,

Thus, the length is $\sqrt{41}$ units, approximately $6.40$ units.

The formula can be used to find polygon side lengths, compare segments for congruence, determine a circle's radius or diameter, classify a triangle, and test whether a point is a specified distance from another point. When comparing lengths, it is often efficient to compare squared distances instead:

This avoids square roots while preserving equality of lengths.

Takeaway: Coordinate differences form the legs of a right triangle, so distance calculations reduce to the Pythagorean theorem.

Averaging coordinates and dividing segments

The midpoint of a segment averages the coordinates of its endpoints:

For endpoints $A(7,-2)$ and $B(9,5)$,

The formula can also be rearranged to find an unknown endpoint. If the midpoint is $M(h,k)$ and one endpoint is $A(x_1,y_1)$, then

For example, with midpoint $(4,1)$ and endpoint $(1,-3)$, the other endpoint is

A midpoint is also useful for geometry problems involving diagonals and perpendicular bisectors.

Takeaway: Average corresponding coordinates to find a midpoint; multiply the midpoint coordinates by $2$ and subtract a known endpoint to recover the other endpoint.

Rate of change and line direction

The slope of a nonvertical line is its rise divided by its run:

For $A(-2,5)$ and $B(4,-1)$,

The line falls one unit for every unit it moves to the right. The sign and value of the slope have these interpretations:

• $m>0$: the line rises from left to right.

• $m<0$: the line falls from left to right.

• $m=0$: the line is horizontal.

• Undefined slope: the line is vertical.

A vertical line has an equation of the form $x=c$, because every point on it has the same $x$-coordinate. A horizontal line has an equation of the form $y=c$, because every point on it has the same $y$-coordinate.

Takeaway: Check the denominator before calculating slope. A zero horizontal change means the line is vertical and its slope is undefined.

Building and comparing line equations

A line can be represented in several forms, each useful in a different situation.

Common line forms

• Slope-intercept form: $y=mx+b$, where $m$ is the slope and $b$ is the $y$-intercept.

• Point-slope form: $y-y_1=m(x-x_1)$, useful when a slope and one point are known.

• Standard form: $Ax+By=C$, usually with integer coefficients, where $A$ and $B$ are not both zero.

To find the equation through $(2,3)$ and $(6,11)$, first calculate

Using point-slope form with $(2,3)$,

After simplifying,

Parallel lines have equal slopes. For example, a line parallel to $y=-3x+4$ has slope $-3$. Through $(2,5)$, its equation is

which simplifies to $y=-3x+11$.

Perpendicular lines have slopes that are negative reciprocals. A line perpendicular to $y=\frac{1}{2}x-7$ has slope $-2$. Through $(4,1)$, its equation is

or $y=-2x+9$.

Takeaway: Find the slope first, choose a line form that matches the given information, and then verify that the resulting equation contains the required point.

Solving line intersections

The intersection of two lines is the point that satisfies both equations. When both equations are solved for $y$, set their right-hand sides equal.

For

and

solve

This gives

Substituting into either equation gives

Therefore, the intersection is $(2,5)$.

The number of solutions describes the geometric relationship:

• One solution means the lines intersect once.

• No solution means the lines are distinct and parallel.

• Infinitely many solutions means both equations describe the same line.

Takeaway: Solve the equations together, then substitute the resulting coordinate into an original equation to check the intersection.

Describing circles with equations

A circle consists of all points at a fixed distance from a center. If its center is $(h,k)$ and its radius is $r$, its standard equation is

For center $(3,-2)$ and radius $5$, substitute carefully:

The signs inside the parentheses are opposite the coordinates of the center. Thus, $(x+4)^2+(y-1)^2=36$ has center $(-4,1)$ and radius $6$.

If the center and a point on the circle are known, calculate the radius with the distance formula:

If the endpoints of a diameter are known, first find the center with the midpoint formula, then find the diameter's length and divide by $2$ to obtain the radius. For endpoints $(-1,-4)$ and $(5,-4)$, the center is $(2,-4)$, the radius is $3$, and the equation is

A circle in general form can be converted to standard form by grouping the coordinate terms and completing the square. For example,

becomes

so its center is $(3,-2)$ and its radius is $5$.

Takeaway: Read the center and radius from standard form, remembering that the signs inside the parentheses are reversed.

Verifying geometric relationships

Coordinate methods turn geometric claims into tests that can be calculated.

Collinearity

Points are collinear when they lie on one line. For three nonvertical points, compare the slope from one point to each of the other two. Equal slopes show that the points lie on the same line. If the line is vertical, check that all points have the same $x$-coordinate.

Congruent segments

Segments are congruent when their lengths are equal. Compare squared distances to avoid unnecessary square roots:

Perpendicular segments

For nonvertical segments, slopes can be multiplied and checked:

A slope-free method uses direction vectors $\vec u=(u_1,u_2)$ and $\vec v=(v_1,v_2)$. Their dot product is

The vectors are perpendicular exactly when

Parallelograms and perpendicular bisectors

A quadrilateral can be shown to be a parallelogram by verifying that opposite sides are parallel or that its diagonals have the same midpoint. The diagonal method also works when sides are vertical.

The perpendicular bisector of a segment passes through its midpoint and is perpendicular to the segment. For $A(1,2)$ and $B(5,4)$, the midpoint is $(3,3)$, and the segment's slope is

The perpendicular slope is $-2$, so the bisector is

or

Takeaway: Select the test that matches the claim: slopes for parallelism and collinearity, distances for congruence, dot products or negative reciprocals for perpendicularity, and midpoints for segment division and diagonal comparisons.

A repeatable problem-solving workflow

A dependable coordinate-geometry solution combines a picture with algebra.

1. Sketch the figure. Label every known point and identify the relationship being tested.

2. Choose the appropriate tool. Use distance, midpoint, slope, a line equation, an intersection method, or a circle equation.

3. Substitute carefully. Keep parentheses around negative coordinates, especially when squaring differences.

4. Simplify and interpret. State what the calculation means geometrically.

5. Check the result. Confirm that the answer satisfies the original conditions and is consistent with the sketch.

The central formulas are

and

Together, these tools allow geometric descriptions to be verified systematically rather than relying only on visual appearance.

• Cartesian plane