Coordinate Geometry: Chapter Review

Vocabulary and notation

Coordinate proof

An argument that represents geometric conditions with coordinates and uses exact calculations to establish a conclusion.

Midpoint

A point on a segment that divides it into equal lengths. Equal endpoint distances alone do not establish a midpoint.

Ordered internal ratio

$AP:PB=u:v$ names the first and remaining lengths from $A$ toward $B$. An internal point lies strictly between distinct endpoints; both parts are positive.

Slope

The signed rise divided by the signed run of a nonvertical line. Horizontal slope is zero; vertical slope is undefined.

Median, altitude, perpendicular bisector

A median joins a vertex to the opposite midpoint. An altitude passes through a vertex perpendicular to the opposite side line. A perpendicular bisector passes through a segment midpoint perpendicular to its line.

Centroid

The common point of a triangle’s medians, lying two thirds of the way from each vertex toward the opposite midpoint.

Locus

The set of all points satisfying a stated geometric condition.

Secant and tangent

For a positive-radius circle, a secant line has two distinct common points; a tangent line has exactly one.

Parabola, focus, directrix, vertex

A parabola consists of points equally distant from a fixed focus and a directrix line not containing it. Its vertex bisects their perpendicular spacing; its axis passes through the focus perpendicular to the directrix.

Formulas and theorems

• Distance

  $$\begin{gathered} AB^2=(x_B-x_A)^2+(y_B-y_A)^2\\ AB=\sqrt{(x_B-x_A)^2+(y_B-y_A)^2} \end{gathered}$$

  Text description

  A slanted segment joins A at the lower left to B at the upper right. Dashed horizontal and vertical legs meet at H with a right angle. These legs measure the coordinate gaps; the solid segment is the straight distance.

  Use when Any two real-coordinate points on perpendicular axes with equal length scales. Coordinate changes are signed; distance is nonnegative. Horizontal, vertical, and coincident endpoints are allowed. Compare squared lengths to order lengths.

  e.g. For endpoints $(-9,5)$ and $(5,-4)$, find the exact length: $\sqrt{277}$ units.

• Midpoint and missing endpoint

  $$\begin{gathered} x_M=\frac{x_A+x_B}{2}\\ y_M=\frac{y_A+y_B}{2}\\ x_B=2x_M-x_A\\ y_B=2y_M-y_A \end{gathered}$$

  Use when Average matching coordinates. The last two rules recover B from A and midpoint M. Coincident endpoints return that same point, without positive half-segments.

  e.g. With $A$ at $(-8,6)$ and midpoint $(-1,-3)$, recover $B$: $(6,-12)$.

• Weighted-coordinate formula

  $$\begin{gathered} \frac{AP}{AB}=\frac{u}{u+v}\\ x_P=\frac{vx_A+ux_B}{u+v}\\ y_P=\frac{vy_A+uy_B}{u+v} \end{gathered}$$

  Text description

  A, P, and B lie in that order on one slanted segment. The first subsegment from A to P is solid blue and labeled u. The remaining subsegment from P to B is dashed red and labeled v. P is strictly between the distinct endpoints.

  Use when Distinct endpoints, positive $u,v$, and ordered internal ratio $AP:PB=u:v$. Apply one fraction to both signed coordinate changes. Equal parts give the midpoint; reversing endpoints reverses the ratio.

• Slope and line equations

  $$\begin{gathered} m=\frac{y_B-y_A}{x_B-x_A}\\ y-y_A=m(x-x_A)\\ \text{vertical: }x=x_A\\ \text{horizontal: }y=y_A \end{gathered}$$

  Use when Use distinct points. The slope quotient requires different first coordinates; point-slope form requires a nonvertical line. Use a fixed first coordinate for a vertical line.

• Parallel-line criterion

  $$\begin{gathered} m_1=m_2 \end{gathered}$$

  Use when For two distinct nonvertical lines, equal slopes are equivalent to parallelism. Equal slopes with a shared point give coincidence. Distinct vertical lines are parallel too.

• Perpendicular-line criterion

  $$\begin{gathered} m_1m_2=-1 \end{gathered}$$

  A counterclockwise quarter-turn sends a nonzero step with signed run $a$ and rise $b$ to run $-b$ and rise $a$.

  Use when The product test is equivalent to perpendicularity for two nonvertical, nonhorizontal lines. Separately, every horizontal line is perpendicular to every vertical line; undefined slope is never multiplied.

• Quadrilateral classification tests

  Diagonals bisecting each other establish a parallelogram. In a parallelogram, one right angle or congruent diagonals establishes a rectangle; congruent adjacent sides establish a rhombus. A rectangle that is also a rhombus is a square.

  Use when Convex quadrilaterals with vertices in boundary order. Match exact coordinate evidence to every hypothesis. Squares also belong to both larger families; every parallelogram is a trapezoid under the inclusive definition.

• Triangle midsegment

  $$\begin{gathered} MN\parallel AB\\ MN=\frac12 AB \end{gathered}$$

  Text description

  M and N are the midpoints of sides AC and BC in a nondegenerate triangle. The single ticks match the two halves of AC; the double ticks match the two halves of BC. The dashed midsegment is parallel to the horizontal base and half its length.

  Use when Nondegenerate triangle ABC; M and N are the midpoints of AC and BC. The midsegment line is distinct from the opposite side line.

• Centroid coordinates and median ratio

  $$\begin{gathered} x_G=\frac{x_A+x_B+x_C}{3}\\ y_G=\frac{y_A+y_B+y_C}{3}\\ AG:GD=2:1 \end{gathered}$$

  Text description

  In triangle ABC, D is the midpoint of BC. The dashed median joins A to D, and G lies two thirds of the way from A to D. The two single ticks on BC mark its equal halves. The vertex-to-centroid portion is twice the remaining portion.

  Use when Noncollinear vertices A, B, C. In the ratio, D is the midpoint of BC and the order starts at vertex A. Average three coordinates separately.

  e.g. For vertices $(-7,5)$, $(2,-7)$, and $(11,8)$, find the centroid: $(2,2)$.

• Polygon perimeter and area

  Perimeter adds outside edge lengths. Area adds pieces with nonoverlapping interiors, or subtracts missing pieces from a containing region. For a triangle and rectangle respectively:

  $$\begin{gathered} A_{\text{triangle}}=\frac12bh\\ A_{\text{rectangle}}=bh \end{gathered}$$

  Text description

  The pentagon has boundary order A, B, C, D, E. The dashed cut EC separates a lower rectangle and an upper triangle. F is the perpendicular foot from D to EC, with a square right-angle mark. The cut and height are internal guides, not outside edges.

  Use when A simple polygon with vertices in boundary order. Pieces must fill the intended region exactly. Use perpendicular base-height pairs and matching units; area uses square units. Internal cuts are absent from perimeter.

• Center-radius equation and real graph cases

  $$\begin{gathered} (x-h)^2+(y-k)^2=r^2 \end{gathered}$$

  Use when Center $(h,k)$ and radius $r>0$. After completing both squares, a positive right side gives a circle; zero gives only $(h,k)$; a negative right side gives no real points.

• Inside, on, or outside a circle

  $$\begin{gathered} d^2=(x-h)^2+(y-k)^2\\ d^2<r^2:\ \text{inside}\\ d^2=r^2:\ \text{on}\\ d^2>r^2:\ \text{outside} \end{gathered}$$

  Text description

  A circle has center C. Point A lies strictly inside it, B is its rightmost boundary point, and D lies outside to the upper right. The radius from C to B provides the boundary distance used to compare the other points.

  Use when A positive-radius circle and a real point at distance d from its center. Compare squared distances exactly, using both coordinate gaps.

• Line-circle intersection count

  A line has two common points when its perpendicular center distance is less than the radius, one when equal, and none when greater.

  Text description

  Three distinct dashed horizontal lines are compared with the same circle. The highest line misses it and has zero common points. The next line touches its top and has one common point. The lowest goes through its center C and has two common points.

  Use when A positive-radius circle and an entire line. Find the perpendicular foot using line equations. A line through the center has distance zero and is a secant. A repeated algebraic root counts once.

• Tangency at a specified point

  $$\begin{gathered} CT\perp\ell \end{gathered}$$

  Text description

  A circle has center C and rightmost point T. The solid contact radius CT is horizontal. A dashed vertical tangent passes through T and has a right-angle mark with the radius. T belongs to both the circle and the line.

  Use when Circle center C, radius $r>0$, and contact T on both the circle and line $\ell$. The line is tangent at T exactly when it is perpendicular to CT. Handle horizontal and vertical directions separately.

• Perpendicular distance to a directrix

  For a horizontal directrix at height c:

  $$\begin{gathered} d=|y-c| \end{gathered}$$

  For a vertical directrix at first coordinate c:

  $$\begin{gathered} d=|x-c| \end{gathered}$$

  Use when A real point at (x,y) and a coordinate-parallel directrix. The foot keeps the coordinate parallel to the line. Use the nonnegative perpendicular gap, not a slanted segment.

• Vertical-axis parabola

  $$\begin{gathered} (x-h)^2=4p(y-k)\\ F=(h,k+p)\\ \text{directrix: }y=k-p\\ \text{axis: }x=h \end{gathered}$$

  Text description

  The downward parabola has vertex V and focus F below it. The dashed horizontal directrix d lies above V. A marked curve point P joins F horizontally and its perpendicular foot H vertically. Matching single ticks mark those equal distances. The corner at H is a right angle.

  Use when Vertex $(h,k)$ and signed $p\ne0$. Positive p opens up; negative p opens down. Vertex-focus distance is $|p|$; focus-directrix spacing is $2|p|$.

• Horizontal-axis parabola

  $$\begin{gathered} (y-k)^2=4p(x-h)\\ F=(h+p,k)\\ \text{directrix: }x=h-p\\ \text{axis: }y=k \end{gathered}$$

  Text description

  The rightward parabola has vertex V and focus F to its right. The dashed vertical directrix d lies to the left. A curve point P joins F vertically and its perpendicular foot H horizontally. Matching single ticks mark the equal distances, and the corner at H is a right angle.

  Use when Vertex $(h,k)$ and signed $p\ne0$. Positive p opens right; negative p opens left. Vertex-focus distance is $|p|$. The focus is off the directrix.

Problem types, step by step

Exam traps

• Trap Replacing distance with a bent route or signed change

  Fix Square complete coordinate differences, add their squares, and take the nonnegative root. Keep radicals exact until rounding is requested.

• Trap Reversing partition weights or ignoring direction

  Fix The first ratio number weights B, the second weights A. Preserve signed changes, check both coordinates, and require a strictly internal fraction.

• Trap Using equal slopes without checking distinctness

  Fix A shared slope and point give coincidence. Distinct vertical lines are parallel with undefined slopes, not infinite slopes.

• Trap Changing only the sign for a perpendicular slope

  Fix Use the negative reciprocal for nonzero defined slopes. Use the separate horizontal-vertical case when a slope is zero or undefined.

• Trap Proving a special drawing or using an incomplete classification test

  Fix Keep arbitrary dimensions and nondegeneracy guards. Establish the required family before applying its converse tests; a square remains both rectangle and rhombus.

• Trap Reading the squared radius as the radius

  Fix Complete both squares and inspect the right-side sign. Only a positive value gives a proper circle; zero is one point and a negative value gives none.

• Trap Proving tangent direction without its contact

  Fix Verify that the contact lies on both the circle and the line. Then verify the perpendicular radius, including horizontal and vertical cases.

• Trap Treating p or four times p as a distance

  Fix Recover signed p from the multiplier. Distances use $|p|$; spacing uses $2|p|$. Exclude zero, and identify which coordinate is squared before choosing the axis.

• Trap Measuring diagonally to the directrix

  Fix Use its perpendicular coordinate gap with an absolute value. Equal squared distances imply equal distances because both are nonnegative.