Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
Page 12 of 19
Each problem type has four distinct practice variants. Open a preview to move among all four.
Translate segment notation into words
Use precise definitions of angle, circle, perpendicular line, parallel line, and line segment from point/line/distance ideas.
Notation can be decoded by asking what the symbol says about extent before reading the letters. We’ll distinguish the bar-over-two-letters form from line and ray notation, then interpret the letters …
Preview problemReplace casual geometry language with precise definitions
Use precise definitions of angle, circle, perpendicular line, parallel line, and line segment from point/line/distance ideas.
Turning casual language into mathematics is a translation job: identify the objects, replace “touch” with an incidence fact, and replace “square corner” with a measurable angle condition. We’ll then connect …
Preview problemFind a midpoint using construction reasoning
Perform formal constructions with compass/straightedge, paper folding, string, reflective devices, or geometry software.
A construction’s auxiliary points are evidence, not automatically the requested point. We’ll use the equal-radius arcs to establish an equidistant locus, identify the line they determine, and then focus on …
Preview problemDetermine whether side lengths can form a constructed triangle
Perform formal constructions with compass/straightedge, paper folding, string, reflective devices, or geometry software.
Triangle construction can be read as a circle-intersection problem. We’ll compare the longest length with the sum of the other two, translate that strictness into the relationship among center distance …
Preview problemIdentify the next step in a construction sequence
Perform formal constructions with compass/straightedge, paper folding, string, reflective devices, or geometry software.
Construction sequences are governed by dependencies: completed arcs create points, and those points determine the next object. We’ll read what the equal-radius intersections guarantee, identify the line uniquely fixed by …
Preview problemDetect an invalid construction step
Perform formal constructions with compass/straightedge, paper folding, string, reflective devices, or geometry software.
Formal construction transfers exact relationships through tool invariants, not through measured approximations. We’ll diagnose which move loses exactness, preserve the original distance as a fixed compass opening, and track how …
Preview problemTransfer a segment length with a compass
Perform formal constructions with compass/straightedge, paper folding, string, reflective devices, or geometry software.
A compass copies a length by carrying the distance itself, so no numerical measurement or rounding is needed. We’ll separate the source segment from the target ray, preserve one compass …
Preview problemConstruct a perpendicular bisector from equal-radius arcs
Perform formal constructions with compass/straightedge, paper folding, string, reflective devices, or geometry software.
The arcs are not decorative; their shared radius encodes equal distances from both segment endpoints. We’ll first ensure the radius is large enough to create two crossings, then use those …
Preview problemConstruct an angle bisector with intersecting arcs
Perform formal constructions with compass/straightedge, paper folding, string, reflective devices, or geometry software.
An angle construction works by building symmetry that can be proved, not by eyeballing a halfway direction. We’ll follow the two preserved-radius steps, record the equal lengths each one creates, …
Preview problemDrop a perpendicular from a point to a line
Perform formal constructions with compass/straightedge, paper folding, string, reflective devices, or geometry software.
To control a line through an outside point, the construction first turns the target line into a chord with two named endpoints. We’ll create two points equidistant from that chord’s …
Preview problemCopy an angle with compass and straightedge
Perform formal constructions with compass/straightedge, paper folding, string, reflective devices, or geometry software.
Copying an angle means reproducing its geometry, and a radius alone does not record how wide the angle opens. We’ll transfer both the source arc radius and the chord between …
Preview problemIdentify a regular polygon construction from its steps
Construct an equilateral triangle, square, and regular hexagon inscribed in a circle.
A regular-polygon construction is best identified from how it spaces vertices around the center. We’ll count the diameter endpoints, translate their spacing into equal central turns, and use the equal-chord …
Preview problemFind the central angle of a regular polygon
Construct an equilateral triangle, square, and regular hexagon inscribed in a circle.
The key distinction is where the angle’s vertex lies: at the polygon or at the circle’s center. We’ll count the equally spaced circle vertices, partition one full turn into that …
Preview problemComplete an inscribed polygon construction sequence
Construct an equilateral triangle, square, and regular hexagon inscribed in a circle.
The next construction step should supply exactly the missing vertices, not jump ahead to connecting points. We’ll treat the existing diameter as one opposite pair, determine the quarter-turn locations required …
Preview problemDiagnose an error in a regular polygon construction
Construct an equilateral triangle, square, and regular hexagon inscribed in a circle.
A regular construction depends on an invariant, so every repeated step should preserve the same geometric quantity. We’ll compare the successive chord steps with the circle’s radius, locate where equal …
Preview problemCompare inscribed regular polygons in the same circle
Construct an equilateral triangle, square, and regular hexagon inscribed in a circle.
Equal circle radii make this a clean chord comparison: the only changing ingredient is the central angle subtended by one side. We’ll compute each polygon’s share of a full turn, …
Preview problemConstruct an equilateral triangle on a segment
Construct an equilateral triangle, square, and regular hexagon inscribed in a circle.
Two equal-radius circles turn one given length into two certified distances without measuring. We’ll use the base as the shared compass opening, interpret a common circle point through the radius …
Preview problemConstruct an inscribed equilateral triangle from six radius steps
Construct an equilateral triangle, square, and regular hexagon inscribed in a circle.
The six radius steps create a circular grid of equal central gaps, but the requested polygon uses only part of that grid. We’ll determine how many basic gaps each of …
Preview problemConstruct an inscribed square with perpendicular diameters
Construct an equilateral triangle, square, and regular hexagon inscribed in a circle.
An inscribed regular figure is controlled by vertex spacing before any sides are drawn. We’ll treat the given diameter as one opposite pair, construct the relationship that produces the second …
Preview problemConstruct an inscribed regular hexagon by stepping the radius
Construct an equilateral triangle, square, and regular hexagon inscribed in a circle.
Stepping a fixed compass opening around a circle links three facts at once: equal chords, equal central angles, and eventual closure. We’ll derive the angle created by one radius-length chord, …
Preview problemWrite a coordinate rule for a translation
Represent transformations as functions on points and compare distance/angle-preserving transformations with non-rigid transformations.
A coordinate transformation rule should encode the same movement for every input point. We’ll translate the horizontal and vertical directions separately into signed coordinate changes, keep each displacement attached to …
Preview problemWrite a coordinate rule for reflection over the x-axis
Represent transformations as functions on points and compare distance/angle-preserving transformations with non-rigid transformations.
A reflection rule comes from the mirror line’s geometry, not from memorizing a random sign change. We’ll identify which coordinate measures position along the axis and which measures signed perpendicular …
Preview problemWrite a coordinate rule for a 90 degree rotation
Represent transformations as functions on points and compare distance/angle-preserving transformations with non-rigid transformations.
For a quarter-turn, the safest route is to track the coordinate directions themselves. We’ll follow where the positive horizontal and vertical unit vectors move, use those images to rebuild a …
Preview problemWrite a coordinate rule for dilation centered at the origin
Represent transformations as functions on points and compare distance/angle-preserving transformations with non-rigid transformations.
A dilation scales the whole position vector from its center, so both coordinates must respond to the same factor. We’ll separate the signed coordinate rule from the nonnegative distance multiplier, …
Preview problemDetermine whether a transformation preserves distance
Represent transformations as functions on points and compare distance/angle-preserving transformations with non-rigid transformations.
Distance preservation concerns the separation between two points, not how far either point travels during the transformation. We’ll apply the rule to a generic pair, subtract their image coordinates, and …
Preview problemDetermine whether a transformation preserves angle measure
Represent transformations as functions on points and compare distance/angle-preserving transformations with non-rigid transformations.
Changing an angle’s location does not necessarily change its measure. We’ll represent its two sides by direction vectors from the vertex, apply the same coordinate map to all three defining …
Preview problemClassify a transformation as rigid or non-rigid
Represent transformations as functions on points and compare distance/angle-preserving transformations with non-rigid transformations.
Rigidity is decided by pairwise distances, not by whether the figure visibly moves or changes position. We’ll track a generic displacement vector through the coordinate rule, extract the segment-length scale …
Preview problemApply a transformation rule to every vertex of a polygon
Represent transformations as functions on points and compare distance/angle-preserving transformations with non-rigid transformations.
A polygon transformation is just the same point function applied repeatedly, with labels and order preserved. We’ll handle each vertex independently, keep the horizontal and vertical changes attached to the …
Preview problemInfer a coordinate transformation rule from point pairs
Represent transformations as functions on points and compare distance/angle-preserving transformations with non-rigid transformations.
One point pair can suggest a rule, but the second pair is what tests whether the pattern is truly consistent. We’ll compute image-minus-source coordinate changes for both mappings, compare those …
Preview problemCompare transformations by their effects on distances and angles
Represent transformations as functions on points and compare distance/angle-preserving transformations with non-rigid transformations.
“Same shape” and “same size” are different claims, so we need separate tests for angles and distances. We’ll find the length multiplier and angle effect for each transformation, then organize …
Preview problemInterpret transformation notation as a function on a point
Represent transformations as functions on points and compare distance/angle-preserving transformations with non-rigid transformations.
Transformation notation behaves like ordinary function notation: one ordered pair is the input, and one ordered pair is the output. We’ll match each input coordinate to its own expression, evaluate …
Preview problemCorrect an error in a transformed image point
Represent transformations as functions on points and compare distance/angle-preserving transformations with non-rigid transformations.
To diagnose a transformation error, return to the original point and write the direction rule before touching the arithmetic. We’ll translate the stated motion into a signed coordinate change, apply …
Preview problemIdentify reflection symmetry lines for a rectangle or square
Describe rotations and reflections that carry rectangles, parallelograms, trapezoids, or regular polygons onto themselves.
A reflection axis must pair every point of the figure with another point of the same figure. We’ll start at the rectangle’s center, test the midlines that exchange equal halves, …
Preview problemIdentify rotational symmetry angles
Describe rotations and reflections that carry rectangles, parallelograms, trapezoids, or regular polygons onto themselves.
Rotational symmetry means the entire outline returns to the same occupied positions, not merely that the figure remains congruent somewhere else. We’ll include the identity case, track what happens to …
Preview problemDistinguish rotational symmetry from reflection symmetry
Describe rotations and reflections that carry rectangles, parallelograms, trapezoids, or regular polygons onto themselves.
A symmetry has to return the whole figure to the same occupied outline, not merely produce a congruent copy somewhere else. We’ll test plausible fold lines by matching corresponding edges, …
Preview problemRecognize reflection symmetry in a trapezoid
Describe rotations and reflections that carry rectangles, parallelograms, trapezoids, or regular polygons onto themselves.
A reflection axis is a fold line that pairs the entire boundary with itself. Since the parallel bases have different lengths, a useful candidate must preserve each base while exchanging …
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