Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
Page 13 of 19
Each problem type has four distinct practice variants. Open a preview to move among all four.
Count and locate reflection symmetry lines of a regular polygon
Describe rotations and reflections that carry rectangles, parallelograms, trapezoids, or regular polygons onto themselves.
Regularity lets us generate reflection axes systematically instead of guessing at lines. Start at a vertex, pass through the center to the midpoint of the opposite side, and then carry …
Preview problemIdentify rotational symmetries of a regular polygon
Describe rotations and reflections that carry rectangles, parallelograms, trapezoids, or regular polygons onto themselves.
A regular polygon returns to itself whenever a turn about its center advances every vertex to another vertex position. Find the equal central-angle step by dividing a full turn by …
Preview problemList all symmetries of a regular polygon
Describe rotations and reflections that carry rectangles, parallelograms, trapezoids, or regular polygons onto themselves.
A complete symmetry inventory has two families that should be counted separately before they are combined. Generate rotations from the equal central-angle step, including the identity, and generate reflections of …
Preview problemDecide whether a proposed symmetry is valid
Describe rotations and reflections that carry rectangles, parallelograms, trapezoids, or regular polygons onto themselves.
A proposed reflection must preserve lengths and vertex adjacencies everywhere, so one local mismatch is enough to disprove it. Track where a long side would land after folding across the …
Preview problemCompare the symmetries of related quadrilaterals
Describe rotations and reflections that carry rectangles, parallelograms, trapezoids, or regular polygons onto themselves.
Compare these figures by viewing a square as a rectangle with the extra condition that adjacent sides are equal. First identify the transformations that preserve any rectangle, then ask which …
Preview problemFind the smallest positive rotation of a regular polygon
Describe rotations and reflections that carry rectangles, parallelograms, trapezoids, or regular polygons onto themselves.
The smallest positive rotational symmetry moves each vertex to the nearest next vertex position. Equal central sectors partition a full turn, so divide the full angle by the number of …
Preview problemComplete all lines of symmetry for a regular polygon
Describe rotations and reflections that carry rectangles, parallelograms, trapezoids, or regular polygons onto themselves.
For an even-sided regular polygon, reflection axes come in two interlaced families. Pair opposite vertices for one family and opposite side midpoints for the other, treating each whole line as …
Preview problemDefine a translation using geometric language
Define rotations, reflections, and translations using geometric language: angles, circles, perpendicular/parallel lines, and segments.
A translation vector describes one shared directed displacement applied to every point, not separate movements for different vertices. Read its first component as horizontal motion and its second as vertical …
Preview problemDefine a rotation using center, angle, and direction
Define rotations, reflections, and translations using geometric language: angles, circles, perpendicular/parallel lines, and segments.
A precise rotation description needs enough information to trace every point’s motion uniquely. Organize that information as a fixed pivot, a directed amount of turn, and the invariant that each …
Preview problemDefine a reflection across a line
Define rotations, reflections, and translations using geometric language: angles, circles, perpendicular/parallel lines, and segments.
A complete reflection definition must handle two kinds of points: those off the mirror and those on it. For an off-line point, characterize its image through the segment joining the …
Preview problemIdentify a rotation from center-distance and angle facts
Define rotations, reflections, and translations using geometric language: angles, circles, perpendicular/parallel lines, and segments.
Rotation facts fit together around one common reference point. Equal distances from that point identify the circle on which the original and image lie, while the directed central angle tells …
Preview problemIdentify a reflection line from a point and its image
Define rotations, reflections, and translations using geometric language: angles, circles, perpendicular/parallel lines, and segments.
A point and its reflected image determine the mirror line without any guesswork. Join the pair, find the segment’s midpoint, and draw the line through that midpoint perpendicular to the …
Preview problemFind the translation vector from a point and its image
Define rotations, reflections, and translations using geometric language: angles, circles, perpendicular/parallel lines, and segments.
A translation vector records the directed change from source to image, so subtraction order matters. Compute image x minus source x and image y minus source y, keep the horizontal …
Preview problemDescribe a transformation with precise defining details
Define rotations, reflections, and translations using geometric language: angles, circles, perpendicular/parallel lines, and segments.
Precise transformation language begins by identifying the motion pattern, then supplying the data that uniquely defines it. When every vertex receives one common directed displacement, encode the horizontal change first …
Preview problemMatch a geometric definition to a transformation type
Define rotations, reflections, and translations using geometric language: angles, circles, perpendicular/parallel lines, and segments.
Transformation types are best recognized by their defining invariants, not by how one point happens to move. Translate the phrase “same distance and direction for every point” into one common …
Preview problemIdentify the missing defining detail in a transformation description
Define rotations, reflections, and translations using geometric language: angles, circles, perpendicular/parallel lines, and segments.
A transformation description is complete only if it produces one unique image for every point. Inventory what a rotation needs—a fixed reference point, a directed angle, and the preserved radius—then …
Preview problemConstruct an image point from the geometric definition of reflection
Define rotations, reflections, and translations using geometric language: angles, circles, perpendicular/parallel lines, and segments.
Construct a reflection from its geometry: the mirror line must perpendicularly bisect the segment from a point to its image. Because this mirror is vertical, preserve the vertical coordinate, measure …
Preview problemCompare fixed points of transformations
Define rotations, reflections, and translations using geometric language: angles, circles, perpendicular/parallel lines, and segments.
A fixed point must land at exactly the same location, which is stronger than merely preserving a distance. Test each transformation with that definition: examine the pivot of the turn, …
Preview problemReplace casual transformation language with precise geometric language
Define rotations, reflections, and translations using geometric language: angles, circles, perpendicular/parallel lines, and segments.
Casual motion words describe an impression, while precise geometry names a transformation and the datum that makes its action unique. Interpret “flip” through the point-image relationship of a mirror, retain …
Preview problemTranslate every vertex of a polygon by the same vector
Draw transformed figures and specify transformation sequences mapping one figure to another.
A polygon translation is one coordinate rule applied uniformly, not a separate decision at each vertex. Convert the vector into horizontal and vertical coordinate changes, apply both components to every …
Preview problemReflect polygon vertices over the x-axis or y-axis
Draw transformed figures and specify transformation sequences mapping one figure to another.
Reflection across a coordinate axis changes only the coordinate measured perpendicular to that axis. For a horizontal mirror, preserve each horizontal position and reverse the signed vertical distance; apply that …
Preview problemReflect vertices over a shifted horizontal or vertical line
Draw transformed figures and specify transformation sequences mapping one figure to another.
A shifted mirror line means ordinary sign-changing shortcuts about the axes no longer apply. Since this line is vertical, keep each vertical coordinate and place the new horizontal coordinate so …
Preview problemRotate vertices about the origin by a standard angle
Draw transformed figures and specify transformation sequences mapping one figure to another.
A quarter-turn about the origin can be handled as one consistent swap-and-sign rule. Derive the sign from the counterclockwise direction rather than memory alone, apply the resulting rule to every …
Preview problemRotate a point about a non-origin center
Draw transformed figures and specify transformation sequences mapping one figure to another.
For a rotation about a non-origin center, work in coordinates relative to that center. Subtract the center to get the radius vector, reverse that vector for a half-turn, and add …
Preview problemDilate a point from a center
Draw transformed figures and specify transformation sequences mapping one figure to another.
A dilation scales the whole center-to-point vector multiplicatively; it does not add the scale factor to coordinates. With the center at the origin, scale both vector components by the same …
Preview problemIdentify a single transformation from a figure description
Draw transformed figures and specify transformation sequences mapping one figure to another.
Identify a transformation from what stays invariant and how corresponding points move. Unchanged size and orientation together with one common displacement rule out scaling, turning, and mirroring; once the motion …
Preview problemApply a two-step transformation sequence in order
Draw transformed figures and specify transformation sequences mapping one figure to another.
A transformation sequence is a pipeline: the output of one rule becomes the input of the next. Record the translated point as an explicit intermediate result before applying the axis …
Preview problemCompare two transformation sequences
Draw transformed figures and specify transformation sequences mapping one figure to another.
Having the same two transformations does not guarantee the same composition, because each second step receives a different intermediate point. Start both paths from the original coordinate, track and label …
Preview problemFind a missing coordinate from a transformation rule
Draw transformed figures and specify transformation sequences mapping one figure to another.
Treat a coordinate rule as two parallel equations, one for each coordinate position. Apply the horizontal component only to the source x-value and the vertical component only to the source …
Preview problemIdentify and correct an incorrectly transformed vertex
Draw transformed figures and specify transformation sequences mapping one figure to another.
Error analysis is most reliable when every listed image is audited against the same rule. Compute image minus source for each labeled pair and compare those difference vectors with the …
Preview problemWrite a transformation sequence using ordered rules
Draw transformed figures and specify transformation sequences mapping one figure to another.
An ordered transformation description should expose both the rule at each stage and the fact that stage two acts on stage one’s output. Encode left and up with signed coordinate …
Preview problemFind a rigid-motion sequence mapping one figure to another
Draw transformed figures and specify transformation sequences mapping one figure to another.
To infer one motion from a restricted family, compare corresponding coordinates before naming the transformation. Compute image minus source for every labeled pair: a constant difference signals one common displacement, …
Preview problemFind a transformation sequence for similar but non-congruent figures
Draw transformed figures and specify transformation sequences mapping one figure to another.
Separate size from position in this two-step mapping. First use a target-to-source side-length ratio to determine the origin-centered dilation and record every intermediate vertex; only then compare intermediate points with …
Preview problemDecide congruence by verifying a translation
Use rigid motions to transform figures and decide whether two figures are congruent.
One matching vertex pair can suggest a translation, but congruence requires the same rigid motion to carry the whole figure. Find a candidate displacement from corresponding points, test it independently …
Preview problemDecide congruence by verifying a reflection
Use rigid motions to transform figures and decide whether two figures are congruent.
A proposed reflection proves congruence only when one mirror rule works for every corresponding point. Translate the vertical-axis mirror into its coordinate effect, audit all labeled pairs under that same …
Preview problemDecide congruence by verifying a rotation
Use rigid motions to transform figures and decide whether two figures are congruent.
Verifying congruence by rotation has two layers: coordinate matching and rigidity. Derive the counterclockwise quarter-turn rule from the geometry, apply it to every preimage vertex, and compare labels in order; …
Preview problem