Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
Page 14 of 22
Each problem type has four distinct practice variants. Open a preview to move among all four.
Verify whether a point lies on a parabola using focus and directrix
Derive the equation of a parabola from a focus and directrix.
Parabola membership is a balance between two specific distances. We’ll compute the point-to-focus distance with coordinate differences, find the shortest perpendicular distance to the directrix rather than a slanted one, …
Preview problemBuild an exact parabola graph specification from focus and directrix
Derive the equation of a parabola from a focus and directrix.
An exact graph specification needs reproducible landmarks, not just a curved sketch. We’ll derive the vertex, signed p, axis, opening, and equation from the focus-directrix geometry, place the latus rectum …
Preview problemFind the vertex of a parabola from focus and directrix
Derive the equation of a parabola from a focus and directrix.
The vertex is not the focus; it is the balance point between the focus and directrix. We’ll project the focus perpendicularly onto the line, take the midpoint of that segment …
Preview problemDetermine the opening direction of a parabola from its focus and directrix
Derive the equation of a parabola from a focus and directrix.
Opening direction depends on where the focus lies relative to the vertex, not on the directrix orientation alone. We’ll use the line’s perpendicular direction for the axis, locate the midpoint …
Preview problemCompare two exact coordinate distances for congruence
Use coordinates to prove simple geometric theorems, including simple circle theorems.
A coordinate congruence proof must measure each requested segment between its own endpoints. We’ll subtract coordinates in a consistent order, form both squared lengths independently, compare those exact values before …
Preview problemCompare two coordinate slopes for parallelism
Use coordinates to prove simple geometric theorems, including simple circle theorems.
Matching slopes establish matching directions, but they do not distinguish parallel lines from one coincident line. We’ll compute both slopes in consistent coordinate order, handle any vertical case before division, …
Preview problemCompare two coordinate slopes for perpendicularity
Use coordinates to prove simple geometric theorems, including simple circle theorems.
Different slopes are not automatically perpendicular; their directions must satisfy a right-angle criterion. We’ll compute each rise-over-run consistently, check first for the horizontal-vertical special case, multiply finite nonzero slopes to …
Preview problemCompare midpoint coordinates to verify bisection
Use coordinates to prove simple geometric theorems, including simple circle theorems.
Intersection alone does not prove bisection; the meeting point must halve both segments. We’ll calculate each midpoint independently by averaging both endpoint coordinates, compare the complete ordered pairs, and use …
Preview problemVerify or reject a coordinate tangent claim
Use coordinates to prove simple geometric theorems, including simple circle theorems.
A tangent claim needs both contact and direction evidence. We’ll verify that the proposed point lies on the circle and candidate line, determine the radius direction from center to contact …
Preview problemVerify a diameter/right-angle claim with squared lengths
Use coordinates to prove simple geometric theorems, including simple circle theorems.
The diameter-angle theorem needs the test point to be on the circle first. We’ll derive the center and radius squared from the diameter endpoints, verify incidence by squared distance, compute …
Preview problemDisprove a claim with a specified coordinate calculation
Use coordinates to prove simple geometric theorems, including simple circle theorems.
Disproving a classification requires only one exact failure of a necessary condition. We’ll name the adjacent-side right-angle requirement for a rectangle, compute both adjacent directions from the coordinates, apply the …
Preview problemChoose the coordinate method needed for a proof goal
Use coordinates to prove simple geometric theorems, including simple circle theorems.
The proof goal determines the coordinate tool: segment congruence asks for equal lengths, not equal directions or shared midpoints. We’ll translate the goal into a length equality, write the endpoint-distance …
Preview problemComplete a coordinate proof from calculated evidence
Use coordinates to prove simple geometric theorems, including simple circle theorems.
A coordinate proof is complete only when the calculated evidence is connected to a theorem with every hypothesis checked. We’ll interpret the two slopes as directions, confirm the lines are …
Preview problemInterpret a coordinate-proof result as a geometric conclusion
Use coordinates to prove simple geometric theorems, including simple circle theorems.
Numerical slope evidence has to be translated according to what slope actually measures: direction. We’ll interpret equality as the same direction, use the stated distinctness to rule out coincidence, identify …
Preview problemFind the midpoint of a segment
Find the point that partitions a directed segment between two points in a given ratio.
A midpoint is halfway in both coordinate directions, so the two coordinates must be averaged separately. We’ll average the endpoint x-values, average the endpoint y-values, combine those results into one …
Preview problemFind the point that partitions a directed segment in a given ratio
Find the point that partitions a directed segment between two points in a given ratio.
A fraction of the way from a named start means scaling the complete directed displacement, not the endpoint coordinates in isolation. We’ll subtract end minus start, multiply both vector components …
Preview problemFind a point that partitions a directed segment in a ratio
Find the point that partitions a directed segment between two points in a given ratio.
An internal ratio first has to become a fraction traveled from the ordered starting endpoint. We’ll add the ratio parts for the total, use the first part as the traveled …
Preview problemInterpret and compute a directed segment partition
Find the point that partitions a directed segment between two points in a given ratio.
Directed partitioning keeps both the endpoint order and the ratio order visible. We’ll translate the first ratio part into a fraction of the whole, subtract in the stated start-to-end direction, …
Preview problemFind a missing endpoint from a partition point
Find the point that partitions a directed segment between two points in a given ratio.
Finding a missing endpoint reverses the partition relation instead of stopping at the known interior point. We’ll solve the midpoint average for the unknown endpoint, substitute coordinatewise, interpret the result …
Preview problemFind the ratio in which a point partitions a segment
Find the point that partitions a directed segment between two points in a given ratio.
To recover a partition ratio, compare the two pieces in their requested order rather than comparing one piece with the whole. We’ll compute the start-to-point and point-to-end displacement vectors, confirm …
Preview problemUse partition points in a real-world context
Find the point that partitions a directed segment between two points in a given ratio.
The context identifies both a starting location and a fraction of the route completed. We’ll translate that language into a directed segment, compute the full coordinate displacement to the destination, …
Preview problemFind the point that is a given fraction of the way from one endpoint to another
Find the point that partitions a directed segment between two points in a given ratio.
Vector partition form keeps the starting point, direction, and fraction in one expression. We’ll write start plus the fraction times end minus start, compute the directed displacement, distribute the scalar …
Preview problemDetermine whether a point lies on a segment before partitioning
Find the point that partitions a directed segment between two points in a given ratio.
Segment membership requires more than lying on the supporting line. We’ll parameterize the finite segment from start to end, solve both coordinate equations for one common parameter, use agreement to …
Preview problemCompare internal and external partition points
Find the point that partitions a directed segment between two points in a given ratio.
Internal and external describe location relative to the endpoints, not whether the two pieces are equal. We’ll record the stated point order, compare it with the between-endpoints and beyond-endpoint definitions, …
Preview problemApply the partition formula with algebraic coordinates
Find the point that partitions a directed segment between two points in a given ratio.
The partition formula works the same way with symbolic coordinates as with numbers. We’ll write the midpoint structure, average the two x-expressions and two y-expressions separately, simplify each coordinate without …
Preview problemFind a line image under dilation, including the invariant case
Verify that dilations send non-center-passing lines to parallel lines and leave center-passing lines unchanged.
Two mapped points determine the exact image line, while center incidence determines whether that line is invariant or merely parallel. We’ll apply the dilation to both coordinates of two source …
Preview problemDerive and verify a parallel image line under dilation
Verify that dilations send non-center-passing lines to parallel lines and leave center-passing lines unchanged.
Direction preservation should be verified from an actual image, not assumed in place of deriving it. We’ll transform the source line’s constant coordinate, confirm the result with two mapped points, …
Preview problemDilate a line that passes through the center
Verify that dilations send non-center-passing lines to parallel lines and leave center-passing lines unchanged.
A line through the dilation center is a special invariant case because every point moves along a ray already contained in that line. We’ll test center incidence, apply the center-passing-line …
Preview problemIdentify a dilation center from corresponding points
Verify that dilations send non-center-passing lines to parallel lines and leave center-passing lines unchanged.
The dilation center lies on every line joining an original point to its image, so two independent correspondence pairs can locate it. We’ll write both correspondence-line equations, intersect them, and …
Preview problemDecide whether one line can be a dilation image of another
Verify that dilations send non-center-passing lines to parallel lines and leave center-passing lines unchanged.
Parallel appearance is necessary for non-center image lines, but feasibility also requires one consistent positive scale. We’ll check whether the source contains the center, derive how its signed offset transforms, …
Preview problemFind the image equation of a line under dilation from the origin
Verify that dilations send non-center-passing lines to parallel lines and leave center-passing lines unchanged.
An origin-centered dilation scales the constant height of every point on a horizontal line while preserving its orientation. We’ll apply the coordinate rule, transform the shared source height, map two …
Preview problemUse dilation to reason about parallel relationships in a figure
Verify that dilations send non-center-passing lines to parallel lines and leave center-passing lines unchanged.
The line-image theorem depends on whether the source side’s supporting line contains the dilation center. We’ll identify the center and source line, use the triangle’s noncollinearity to settle incidence, apply …
Preview problemDistinguish dilation effects on infinite lines and finite segments
Verify that dilations send non-center-passing lines to parallel lines and leave center-passing lines unchanged.
Set invariance does not mean every point stays fixed, and an infinite line is not a finite segment with a total length. We’ll parameterize the center-passing line, see how dilation …
Preview problemFind the image length of a segment after a dilation
Verify that dilations scale line segments by the dilation scale factor.
A dilation scale factor is a multiplicative length ratio, regardless of the segment’s direction or the center’s location. We’ll identify the original length and positive factor, multiply them in the …
Preview problemVerify segment scaling using the distance formula
Verify that dilations scale line segments by the dilation scale factor.
Coordinate scaling should agree with actual segment-length scaling. We’ll compute the preimage distance from both coordinate changes, compute the image distance independently, form the image-to-preimage ratio in the correct order, …
Preview problemFind the image length of a segment after a dilation
Verify that dilations scale line segments by the dilation scale factor.
Segment length is a one-dimensional measure, so a dilation multiplies it by the absolute scale factor. We’ll source the original length and factor, apply the relationship in the forward direction, …
Preview problem