298 core topics
+ 31 prerequisite topics taught
as needed · approximately 98 hours of instruction
including spaced review
An adaptive diagnostic (up to
40 questions) places the student on the course's knowledge
graph - topics already known are credited, and instruction begins exactly
at the learning frontier. Every topic is taught with a worked-example
lesson and auto-graded practice; a topic is mastered at
75%+ and then maintained through spaced reviews on an
expanding schedule. Mixed checks follow every 6 lessons;
each unit ends with a 12-item quiz, and course-wide assessments appear at
25%, 50%, 75%, and 100% mastery. Prerequisite gaps below the course are detected and taught rather
than skipped, so completion certifies the whole tower, not just the top.
| The Triangle Inequality
[M] |
Any two sides together must outreach the third. |
| Isosceles & Equilateral Triangles
[M] |
Equal sides sit opposite equal angles. |
| The Exterior Angle Theorem
[M] |
An exterior angle equals the two far-away interior angles combined. |
| Congruence Criteria (SSS · SAS · ASA · AAS)
[M] |
Which marked parts force two triangles to match exactly. |
| Using Congruence (CPCTC)
[M] |
Once triangles are congruent, every matching part is equal. |
| The Midsegment Theorem
[M] |
The segment joining two midpoints is half the far side. |
| Parallelogram Properties
[M] |
Opposite sides equal, consecutive angles supplementary, diagonals bisect. |
| Special Quadrilaterals
[M] |
Rhombus, rectangle, square, trapezoid - by their defining properties. |
| The Inscribed Angle Theorem
[M] |
An inscribed angle is half the central angle on the same arc. |
| Tangent Lines to Circles
[M] |
A tangent meets its radius at a right angle. |
| Intersecting Chords
[M] |
Crossing chords cut each other into equal products. |
| Coordinate Geometry Proofs
[M] |
Prove geometric facts with slopes, distances, and midpoints. |
| Hypotenuse, Opposite & Adjacent
[E] |
Name the three sides of a right triangle relative to a chosen angle. |
| The Sine Ratio
[E] |
Sine is opposite over hypotenuse - read it straight off the triangle. |
| The Cosine Ratio
[E] |
Cosine is adjacent over hypotenuse - the leg that touches the angle. |
| The Tangent Ratio
[E] |
Tangent is opposite over adjacent - the only ratio with no hypotenuse. |
| Choosing the Right Ratio
[M] |
Match the two sides in play to sine, cosine, or tangent. |
| Pythagoras, Then the Ratio
[M] |
When only two sides are given, the Pythagorean theorem supplies the third. |
| Finding a Side from a Given Ratio
[M] |
Multiply the known side by the given ratio to reach the unknown one. |
| When the Unknown Is on the Bottom
[M] |
Divide by the ratio when the unknown side sits in the denominator. |
| Exact Values: the 45-45-90 Triangle
[M] |
Half a square: legs equal, hypotenuse √2 times a leg. |
| Exact Values: 30° and 60°
[M] |
Half an equilateral triangle gives every 30° and 60° value exactly. |
| Sides of the 30-60-90 Triangle
[M] |
Short leg x, long leg x√3, hypotenuse 2x - always in that pattern. |
| Angle of Elevation Problems
[H] |
Ground distance, height, and line of sight form a right triangle. |
| Cofunctions: sin x = cos (90 − x)
[M] |
Complementary angles trade sine and cosine - same triangle, other corner. |
| Translations by a Vector
[E] |
Slide every point the same amount: add the vector to the coordinates. |
| Finding the Translation
[E] |
Image minus preimage recovers the vector; subtract it to go back. |
| Reflections over the Axes
[E] |
The mirror line's own coordinate stays; the other flips sign. |
| Reflections over y = x
[E] |
Over y = x the coordinates swap; over y = −x they swap and negate. |
| Rotations of 90° about the Origin
[M] |
Quarter turns swap the coordinates and flip one sign. |
| Rotations of 180° and 270°
[M] |
A half turn negates both coordinates; 270° is a quarter turn the other way. |
| Identifying a Transformation
[M] |
Read the coordinate rule off a preimage-image pair. |
| Composing Transformations
[M] |
Apply the first rule, then feed its output into the second. |
| Dilations with Fractional Scale Factors
[M] |
Multiply every coordinate by the scale factor - even when it is a fraction. |
| Finding the Scale Factor
[M] |
Scale factor = image measurement divided by original measurement. |
| Rotational Symmetry
[E] |
Order n means n matching positions per full turn - every 360/n degrees. |
| Congruent or Similar?
[M] |
Rigid motions keep congruence; any leftover dilation only keeps similarity. |
| Copying a Segment
[E] |
A compass transfers a length exactly, so a copied segment matches the original. |
| Copying an Angle
[E] |
Transferring an angle's arc and chord reproduces its measure exactly. |
| Bisecting a Segment
[E] |
Equal arcs from both endpoints locate the midpoint and the perpendicular bisector. |
| Bisecting an Angle
[M] |
The angle bisector cuts an angle into two congruent halves. |
| The Perpendicular Bisector Property
[M] |
A point is equidistant from two endpoints exactly when it lies on their perpendicular bisector. |
| Constructing a Perpendicular
[M] |
Dropping or raising a perpendicular is a perpendicular-bisector construction that yields right angles. |
| Constructing a Parallel
[M] |
Copying a transversal's angle makes equal corresponding angles, forcing the lines parallel. |
| Identifying a Construction
[E] |
Read a sequence of compass-and-straightedge steps and name the construction. |
| Why Constructions Work
[M] |
The validity of each construction rests on congruent triangles and equidistance. |
| Inscribing a Regular Hexagon
[M] |
Stepping the radius around a circle marks six points - a regular hexagon of side equal to the radius. |
| Inscribing an Equilateral Triangle
[M] |
Joining every other of the six hexagon points gives an inscribed equilateral triangle. |
| Loci: Sets of Equidistant Points
[M] |
A locus is the full set of points meeting a distance condition - a bisector, a circle, or parallels. |
| Points of Concurrency
[M] |
The centroid cuts each median 2:1, and the circumcenter is equidistant from all three vertices. |
| The Midpoint of a Segment
[E] |
Average the x's and average the y's to land exactly in the middle. |
| Finding the Other Endpoint
[M] |
Reverse the midpoint formula: the midpoint is halfway, so double and back off. |
| Distance Between Two Points
[M] |
The distance is the hypotenuse of the run-and-rise right triangle. |
| Perimeter of a Polygon on the Grid
[M] |
Walk the vertices in order, measure every side, and add the lengths. |
| Area of an Axis-Aligned Rectangle
[M] |
Width times height, where each dimension is a coordinate difference. |
| Area of a Triangle from Its Vertices
[M] |
One side as base, the perpendicular distance to the opposite vertex as height. |
| Partitioning a Segment in a Ratio
[M] |
The dividing point sits m/(m+n) of the way from A toward B. |
| A Fraction of the Way Along
[M] |
Add k times the whole displacement to the starting point. |
| A Line Parallel to a Given Line
[M] |
Parallel lines share a slope; solve for the new intercept from the point. |
| A Line Perpendicular to a Given Line
[M] |
Flip and negate the slope, then fit the intercept to the point. |
| Is the Triangle Right? (Slopes)
[M] |
Two sides meet at a right angle exactly when their slopes multiply to −1. |
| Right, Acute or Obtuse (Pythagoras' Converse)
[M] |
Compare the longest side squared with the sum of the other two squares. |
| Completing a Parallelogram
[H] |
A quadrilateral is a parallelogram exactly when its diagonals share a midpoint. |
| What Fraction of the Circle?
[E] |
A central angle takes the fraction theta/360 of the whole circle. |
| Arc Length as a Piece of the Circumference
[M] |
Arc length is (theta/360) of the circumference 2*pi*r. |
| Sector Area as a Slice of the Circle
[M] |
Sector area is (theta/360) of the circle area pi*r^2. |
| From a Fraction Back to the Angle
[M] |
If an arc is a given fraction of the circle, the angle is that fraction of 360°. |
| Degrees to Radians
[M] |
A radian sweeps one radius of arc; multiply degrees by pi/180. |
| Radians to Degrees
[M] |
Multiply a radian measure by 180/pi to get degrees. |
| Arc Length with Radians: s = r*theta
[M] |
In radians the arc length is simply the radius times the angle. |
| Sector Area with Radians: A = ½r²θ
[M] |
With theta in radians a sector's area is one half r squared theta. |
| Finding the Radius from an Arc
[M] |
Invert the arc-length formula to recover the radius. |
| Inscribed & Central Angles on One Arc
[M] |
The central angle equals its arc; the inscribed angle is half of it. |
| The Tangent-Chord Angle
[M] |
A tangent-chord angle is half the arc it cuts off. |
| Central Angle from an Arc Length
[M] |
Compare the arc to the whole circumference to recover the angle. |
| Area with Two Sides and an Angle
[M] |
Two sides and the angle between them give the area directly. |
| Area Backwards: Find a Missing Side
[M] |
Turn the area formula around to recover a side length. |
| Law of Sines: Finding a Side
[M] |
Each side over the sine of its opposite angle stays constant. |
| Law of Sines: Finding an Angle
[M] |
Solve the proportion for a sine, then read off the special angle. |
| Law of Cosines: The Third Side
[H] |
c² = a² + b² − 2ab cos C reaches the side the Law of Sines can't. |
| Law of Cosines: Finding the Angle
[H] |
Three sides pin down every angle through its cosine. |
| Classifying a Triangle by Its Sides
[M] |
The sign of a² + b² − c² tells acute from right from obtuse. |
| Choosing Sines vs. Cosines
[M] |
The marked parts decide which law does the job. |
| Multi-Step Angle of Elevation
[H] |
Two sight lines to the same top pin down an unknown height. |
| Multi-Step Angle of Depression
[H] |
One height, two depression angles, and the gap between the targets. |
| Law of Sines in the Field
[M] |
Surveying and navigation triangles solved with one clean proportion. |
| SAS Area in Context
[M] |
Real plots and gardens measured from two sides and their angle. |
| Probability on a Segment
[E] |
A point on a segment lands in a region with probability length over length. |
| Length Models in Context
[M] |
Waiting times and positions are segment probabilities in disguise. |
| Probability by Area
[M] |
A dart on a region lands in a shape with probability area over area. |
| Triangular Targets
[M] |
Same ratio idea, but the favorable area is half base times height. |
| Landing Inside an Inscribed Circle
[M] |
Circle area over rectangle area keeps pi symbolic - report its coefficient. |
| Rings and Concentric Circles
[M] |
When both regions are circles, pi cancels and the answer is a clean fraction. |
| Composite and L-Shaped Regions
[M] |
Find the favorable area by adding or subtracting rectangles, then divide. |
| Population as Area Density
[M] |
Population equals people-per-area times area - density is a rate over region. |
| Comparing Population Densities
[M] |
Denser means more people per square mile, so compare population over area. |
| Choosing the Right Units
[E] |
Track units through a model: area is squared, and a probability is unitless. |
| Designing a Region to a Constraint
[M] |
Work backward from a required area to the dimension that meets it. |
| Expected Value from Geometric Probability
[H] |
Weigh each payout by its area-probability and add up the pieces. |
| AA Similarity: Solving for a Side
[H] |
Two matching angle pairs force similarity; then a proportion finds any side. |
| Similarity Criteria (AA · SAS~ · SSS~)
[M] |
Which given ratios and angles force two triangles to be similar. |
| Congruent Triangles: Full Solves
[H] |
Match corresponding parts, solve for the unknown, then total the sides. |
| The Side-Splitter Theorem
[H] |
A line parallel to one side cuts the other two sides in the same ratio. |
| Geometric Means in Right Triangles
[H] |
The altitude to the hypotenuse creates three similar triangles. |
| Similar Figures: Lengths vs Areas
[H] |
Scale lengths by k and every area scales by k squared. |
| Elevation with 30°, 45° and 60°
[H] |
Special angles give exact heights - no calculator, no rounding. |
| Angle of Depression Problems
[H] |
Looking down makes the same right triangle as looking up. |
| Slopes, Ramps & Road Grades
[H] |
A slope ratio or percent grade is the tangent of the incline angle. |
| Tangent-Secant Power of a Point
[H] |
tangent squared equals external part times whole secant. |
| Two Secants from an External Point
[H] |
external times whole matches for every secant from the same point. |
| Cyclic Quadrilaterals
[H] |
Opposite angles of an inscribed quadrilateral add to 180 degrees. |
| The Angle Between Two Chords
[H] |
An inside angle is half the sum of its two intercepted arcs. |
| Angles Formed Outside a Circle
[H] |
An outside angle is half the difference of the far and near arcs. |
| Coordinate Proof: Classifying a Triangle
[H] |
Squared distances settle both the sides and the angles, exactly. |
| Coordinate Proof: Collinear Points
[H] |
Three points are collinear exactly when the slopes agree. |
| Coordinate Proof: Diagonals Bisect
[H] |
A parallelogram is proved by one shared midpoint computation. |
| Coordinate Proof: Trapezoid Midsegment
[H] |
Midpoints of the legs join into a segment averaging the two bases. |
| Finding the Radius from a Sector
[H] |
Invert the sector-area formula to recover the radius. |
| Area of a Circular Segment
[H] |
Segment equals sector minus the triangle on the chord. |
| Slices of a Ring
[H] |
A slice of an annulus takes theta/360 of the ring's area. |
| Density, Mass & Cost Modeling
[H] |
Volume feeds density feeds cost - a chain of rates. |
| Filling a Tank: Volume and Rate
[H] |
Time to fill equals the volume divided by the flow rate. |
| Cross-Sections of Solids
[M] |
What flat shape appears when a plane slices a solid. |
| Dimensions of a Solid of Revolution
[H] |
The axis side becomes the height; the swinging side becomes the radius. |
| The Smallest Admissible Third Side
[H] |
The third side must exceed the difference of the other two, so the smallest whole-number value sits just above that difference. |
| Which Three Lengths Build a Triangle
[H] |
Three lengths form a triangle exactly when the two shorter ones together exceed the longest. |
| Bounds on the Whole Perimeter
[H] |
Adding the two known sides to each end of the third-side range bounds the perimeter of the triangle. |
| Triangle Inequality with a Variable Side
[H] |
Writing all three inequalities in terms of the variable pins down the range of admissible values. |
| Ordering Sides by Their Opposite Angles
[H] |
In any triangle the longer side lies opposite the larger angle, so ordering the angles orders the sides. |
| Exterior Angles of an Isosceles Triangle
[H] |
Equal base angles turn the exterior angle at a base vertex into 90 degrees plus half the apex angle. |
| Two Exterior Angles at Once
[H] |
One exterior angle at each vertex of a triangle totals 360 degrees, which links any two of them to the third. |
| The Exterior Angle Inequality
[H] |
Because an exterior angle equals the sum of two positive remote interior angles, it strictly exceeds each of them. |
| The Midsegment Triangle's Perimeter
[H] |
Joining the three midpoints of a triangle gives a triangle whose perimeter is exactly half the original. |
| Area on Either Side of a Midsegment
[H] |
A midsegment cuts off a triangle similar at ratio 1 to 2, so it takes one quarter of the area and leaves three quarters. |
| Midsegments on the Coordinate Plane
[H] |
On a grid a midsegment is found by averaging coordinates, and its length is half the distance across the side it parallels. |
| Median Pieces at the Centroid
[H] |
The centroid cuts each median so the vertex piece is twice the other, making the whole median three halves of the vertex piece. |
| Recovering a Vertex from the Centroid
[H] |
Because the centroid is the average of the three vertices, any missing vertex is three times the centroid minus the other two. |
| The Length of a Median
[H] |
A median's length is the distance from a vertex to the average of the other two vertices. |
| Medians and Equal Areas
[H] |
A median halves a triangle's area, all three medians cut it into six equal pieces, and each vertex-centroid triangle takes one third. |
| Altitudes from the Area
[H] |
Every side of a triangle pairs with its own altitude through the same area, so one base-height pair determines all the others. |
| Locating the Orthocenter
[H] |
The orthocenter is the common point of the three altitudes, found by intersecting two perpendicular-to-a-side lines through the opposite vertices. |
| Where Each Triangle Center Lies
[H] |
The centroid and incenter always lie inside, while the circumcenter and orthocenter move outside for an obtuse triangle. |
| The Circumcenter of a Right Triangle
[H] |
In a right triangle the circumcenter is the midpoint of the hypotenuse, so the circumradius is half the hypotenuse. |
| The Circumcenter from Three Vertices
[H] |
The circumcenter is the single point equidistant from all three vertices, located by intersecting two perpendicular bisectors. |
| Equidistance on a Perpendicular Bisector
[H] |
A point on the perpendicular bisector of a segment is equidistant from its endpoints, which turns two expressions into one equation. |
| The Angle at the Incenter
[H] |
The angle subtended at the incenter by one side equals 90 degrees plus half the opposite angle. |
| The Inradius from Area and Semiperimeter
[H] |
The incircle's radius equals the triangle's area divided by its semiperimeter. |
| Between the Bisector and the Altitude
[H] |
The angle between the bisector and the altitude from one vertex equals half the difference of the other two angles. |
| Writing the Similarity Statement
[H] |
A similarity statement is correct only when matching positions in the two names hold equal angles. |
| Completing an SAS Similarity
[H] |
With the included angles equal, similarity holds exactly when the two pairs of sides around them share one ratio. |
| SSS Similarity and the Scale Factor
[H] |
When all three side pairs share one ratio the triangles are similar, and that ratio is the scale factor. |
| Proportions with the Unknown on Both Sides
[H] |
Cross multiplying a proportion from similar triangles turns a repeated unknown into a linear equation. |
| Corresponding Altitudes, Medians and Bisectors
[H] |
In similar triangles every corresponding length, including altitudes medians and angle bisectors, shares the side ratio. |
| Is the Segment Parallel to the Side?
[H] |
A segment joining two sides of a triangle is parallel to the third side exactly when it divides those sides in the same ratio. |
| Three Parallel Lines Cutting Two Transversals
[H] |
Three parallel lines cut off segments in the same ratio on every transversal that crosses them. |
| The Angle-Bisector Proportionality Theorem
[H] |
An angle bisector divides the opposite side into two pieces proportional to the two adjacent sides. |
| Angle Bisectors and the Whole Triangle
[H] |
Combining the bisector ratio with the perimeter recovers sides that no single measurement gives directly. |
| The Mean Proportional
[H] |
The geometric mean of two numbers is the square root of their product, the value that makes a proportion with itself in both middle positions. |
| Working Backwards from a Leg or an Altitude
[H] |
The three similar triangles made by the altitude let any two known lengths on the hypotenuse recover the rest. |
| Area and Perimeter from the Hypotenuse Split
[H] |
The two hypotenuse segments determine the altitude and both legs, and therefore the whole triangle's area and perimeter. |
| Scale Factor and Perimeter
[H] |
Perimeter is a length, so it scales by the scale factor itself rather than by its square. |
| Material and Cost Under Scaling
[H] |
Anything proportional to area, such as cloth paint or coating cost, scales by the square of the length ratio. |
| Enlargements Run Forwards and Backwards
[H] |
An enlargement multiplies each dimension by the scale factor and the area by its square, so an area ratio recovers the factor by a square root. |
| Indirect Measurement with Shadows
[H] |
Objects and their shadows at the same moment form similar right triangles, so height compares to shadow in a fixed ratio. |
| The Mirror Method
[H] |
A mirror on the ground reflects at equal angles, creating two similar right triangles that relate eye height to object height. |
| Sighting Across a Gap
[H] |
Two sight lines crossing at a stake create vertical angles and similar triangles, so an unreachable width follows from measurable baselines. |
| Arc Addition Around a Full Circle
[H] |
The arcs cut by points on a circle add to exactly 360 degrees. |
| Inscribed Angles with Algebraic Arcs
[H] |
Set the inscribed angle equal to half its arc and solve for the variable. |
| Two Inscribed Angles on One Chord
[H] |
Inscribed angles standing on the same arc are congruent. |
| The Angle in a Semicircle
[H] |
An angle inscribed in a semicircle is a right angle. |
| Angles of an Inscribed Triangle
[H] |
Each angle of an inscribed triangle is half the arc opposite it. |
| Arcs Given as a Ratio
[H] |
Share 360 degrees among the ratio parts, then read off the arc or angle. |
| Where the Vertex Sits Decides the Rule
[H] |
Center, on, inside or outside the circle selects which arc rule to use. |
| Tangent-Chord Angles Run Backwards
[H] |
A tangent-chord angle is half the arc it closes off, from either side. |
| A Tangent and a Secant Outside a Circle
[H] |
The external angle is half the difference of the far and near arcs. |
| Cyclic Quadrilateral Angles from Arcs
[H] |
Each angle of an inscribed quadrilateral is half the two arcs across from it. |
| Tangent Length from an External Point
[H] |
Radius, tangent segment and center distance form a right triangle. |
| Congruent Tangents and Perimeters
[H] |
Two tangent segments drawn from one external point are congruent. |
| A Radius Perpendicular to a Chord
[H] |
The perpendicular from the center bisects a chord and builds a right triangle. |
| Comparing Chords and Their Distances
[H] |
Chords equally far from the center are congruent, and longer chords sit closer. |
| The Power of a Point
[H] |
The number d squared minus r squared measures every product through a point. |
| Chord Products That Need a Quadratic
[H] |
Intersecting-chord products become quadratic equations when a piece is unknown. |
| The Perimeter of a Sector
[H] |
A sector's perimeter is its arc length plus two radii. |
| The Sector That Rolls into a Cone
[H] |
A cone's net is a sector whose arc becomes the base circumference. |
| Rolling Wheels and Turning Arcs
[H] |
A rolling wheel advances by the arc length that touches the ground. |
| Writing a Circle's Equation
[H] |
A center and one radius or one point on the circle fix the equation. |
| Reading Center and Radius Back
[H] |
Standard form displays the center and the square of the radius. |
| Circles by Completing the Square
[H] |
Completing the square converts general form to center-radius form. |
| A Circle from the Ends of a Diameter
[H] |
The center is the diameter's midpoint and the radius is half its length. |
| Inside, On, or Outside a Circle
[H] |
Compare the squared distance from the center with the squared radius. |
| Circles Tangent to the Axes
[H] |
Tangency to a line means the distance from the center equals the radius. |
| Pyramid Volume, Run Backwards
[H] |
One third base times height solves for whichever measurement is missing. |
| Slant Height, Radius and Height
[H] |
A cone's radius, height and slant height form a right triangle. |
| Surface Area of a Cone
[H] |
Lateral area is pi r l and the base adds pi r squared. |
| Surface Area of a Square Pyramid
[H] |
Four triangles on a square base: s squared plus two s times the slant height. |
| Sphere Surface Area and Volume Together
[H] |
Four pi r squared and four thirds pi r cubed share the same radius. |
| Surface Area of a Triangular Prism
[H] |
Two triangular ends plus the perimeter of the base times the length. |
| Surface Area of a Composite Solid
[H] |
Add only the faces that remain exposed after the pieces are joined. |
| A Missing Dimension from a Volume
[H] |
Solve the volume formula for the radius or height instead of the volume. |
| Volume of a Frustum
[H] |
A frustum is a solid with its top cut off, measured by subtraction or by one formula. |
| Which Solid Holds the Most
[H] |
Compare volumes by their pi coefficients, not by how large the solids look. |
| Troughs: Prisms Lying on Their Side
[H] |
A trough's volume is its cross-sectional area times its length. |
| The Area of a Cross-Section
[H] |
Find the slice's dimensions first, then apply the flat-shape area formula. |
| Slicing a Cube
[H] |
A cube's cross-sections range from squares and triangles to a regular hexagon. |
| From a Ratio Back to the Scale Factor
[H] |
Square root an area ratio and cube root a volume ratio to recover lengths. |
| Scaling in Context: Paint, Mass and Capacity
[H] |
Coverage scales with the square of length and mass or capacity with the cube. |
| Density with Cylinders, Balls and Pipes
[H] |
Mass equals density times volume, even when the volume carries a factor of pi. |
| Packing, Filling and Counting Loads
[H] |
Volume division counts loads, but packing solid pieces counts along each edge. |
| Coating Cost from Surface Area
[H] |
Cost follows the area actually covered, rounded up to whole units of product. |
| Classifying a Quadrilateral from Its Vertices
[H] |
Slopes decide which sides are parallel or perpendicular, and squared distances decide which sides are congruent. |
| Diagonals of a Coordinate Quadrilateral
[H] |
A diagonal is just the distance between two opposite vertices, so the distance formula measures and compares diagonals. |
| Area of a Quadrilateral from Its Vertices
[H] |
The coordinate area formula pairs each vertex with the next and halves the alternating sum of the cross products. |
| Side Lengths in Simplest Radical Form
[H] |
A coordinate length is a square root, and pulling out perfect-square factors leaves it exact instead of rounded. |
| Completing an Isosceles Trapezoid
[H] |
An isosceles trapezoid is symmetric about the perpendicular bisector of its bases, so the two top vertices are inset equally from the ends. |
| A Point Equidistant from Two Others
[H] |
Setting the two squared distances equal turns the equidistance condition into a linear equation. |
| Completing a Rectangle or a Square
[H] |
Opposite sides of a rectangle are equal vectors, and a square's next side is that vector turned a quarter turn. |
| The Equation of a Perpendicular Bisector
[H] |
The perpendicular bisector passes through the midpoint with slope the opposite reciprocal of the segment's slope. |
| Parallel and Perpendicular Lines from Standard Form
[H] |
Keeping the coefficients of a standard-form line gives a parallel line, while swapping them and changing one sign gives a perpendicular one. |
| Where a Perpendicular Meets a Line
[H] |
The foot of a perpendicular is the intersection of the given line with the perpendicular line through the outside point. |
| Distance from a Point to a Line
[H] |
The distance from a point to a line is measured along the perpendicular, and the standard-form coefficients compute it directly. |
| Recovering the Ratio from the Point
[H] |
Comparing the run from A to P with the run from A to B recovers the fraction and hence the ratio of the two pieces. |
| Finding an Endpoint from a Partition Ratio
[H] |
If P cuts AB in the ratio m to n, then the step from A to P is m/(m+n) of the whole step, which recovers either endpoint. |
| Reflections over x = a and y = b
[H] |
A mirror line halfway between a point and its image gives the rule x maps to 2a minus x for a vertical mirror. |
| Finding the Line of Reflection
[H] |
The mirror line is the perpendicular bisector of the segment joining any point to its image. |
| Two Reflections over Parallel Lines
[H] |
Reflecting in two parallel mirrors is a translation perpendicular to them through twice the distance between them. |
| Two Reflections over Intersecting Lines
[H] |
Reflecting in two intersecting mirrors is a rotation about their intersection through twice the angle between them. |
| Glide Reflections
[H] |
A glide reflection is a translation along a line followed by a reflection in that same line, and the two steps commute. |
| Rotating about a Point Other than the Origin
[H] |
Shift the center to the origin, apply the origin rotation rule, then shift back. |
| A Sequence That Maps One Figure onto Another
[H] |
Two congruent figures are related by a sequence of rigid motions, and testing every vertex decides which sequence works. |
| Is the Rule a Rigid Motion?
[H] |
A rigid motion preserves every distance, a similarity scales all distances by one factor, and anything else distorts the figure. |
| What a Transformation Preserves
[H] |
Rigid motions preserve length and angle, dilations preserve angle and slope but not length, and reflections reverse orientation. |
| Counting Lines of Symmetry
[H] |
A line of symmetry is a mirror that maps the figure exactly onto itself, and each candidate must send every vertex to a vertex. |
| The Equation of a Line of Symmetry
[H] |
An axis of symmetry passes through the midpoints of the segments joining mirror-image vertices. |
| Rotations That Carry a Regular Polygon onto Itself
[H] |
A regular n-gon returns to itself exactly at the multiples of 360/n degrees, and it has n lines of symmetry as well. |
| Dilations Centered Away from the Origin
[H] |
A dilation multiplies the vector from the center to the point by the scale factor, so the center behaves like a temporary origin. |
| How a Dilation Changes Slope and Length
[H] |
A dilation leaves the slope of every segment unchanged while multiplying every length by the scale factor. |
| Recovering the Center of a Dilation
[H] |
Because a point, the center and the image are collinear, the center is the point that solves the image equals center plus k times point minus center. |
| The Image of a Line under a Dilation
[H] |
A dilation sends a line to a parallel line, so only its intercept moves and the slope is untouched. |
| Similarity Transformations Mapping One Figure onto Another
[H] |
Two similar figures are related by a dilation followed by a rigid motion, and the ratio of corresponding lengths gives the scale factor. |
| From One Trig Ratio to Another
[H] |
One ratio fixes two sides of a right triangle, and the Pythagorean theorem supplies the third so any other ratio follows. |
| Recovering the Angle from an Exact Ratio
[H] |
The three special ratios run backwards: a ratio of 1/2, sqrt(2)/2 or sqrt(3) identifies the angle without a calculator. |
| Base Angles, Heights and Legs
[H] |
Dropping perpendiculars from the shorter base turns a trapezoid into a rectangle flanked by two right triangles. |
| 30-60-90 Triangles: Exact Sides and Areas
[H] |
In a 30-60-90 triangle the sides run short leg, short leg times sqrt(3), and twice the short leg, which makes every area exact. |
| 45-45-90 Triangles: Squares and Diagonals
[H] |
A square's diagonal is its side times sqrt(2), so the diagonal determines the side, the perimeter and the area exactly. |
| Equilateral Triangles via the 30-60-90 Split
[H] |
An altitude splits an equilateral triangle into two 30-60-90 triangles, giving altitude s times sqrt(3) over 2 and area s squared times sqrt(3) over 4. |
| Apothem and Area of a Regular Hexagon
[H] |
A regular hexagon splits into six equilateral triangles, so its apothem is half the side times sqrt(3) and its area is six of those triangles. |
| Elevation Measured from Eye Level
[H] |
The right triangle of a sighting starts at eye level, so the observer's eye height must be added back to the computed rise. |
| Two Sightings from One Height
[H] |
Two depression angles from the same height give two horizontal distances, and their difference is the gap between the targets. |
| Slope and the Angle a Line Makes
[H] |
The slope of a line equals the tangent of the angle it makes with the positive x-axis, which links steepness to angle measure. |
| How the Ratios Behave as the Angle Grows
[H] |
As an acute angle grows the sine increases toward 1, the cosine decreases toward 0, and the tangent increases without bound. |
| Finding an Angle of Depression from Measurements
[H] |
A height and a horizontal distance determine the tangent of the depression angle, and the special ratios name that angle exactly. |