Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
Page 15 of 22
Each problem type has four distinct practice variants. Open a preview to move among all four.
Find the original segment length from the image length and dilation scale factor
Verify that dilations scale line segments by the dilation scale factor.
Recovering a preimage length means undoing the forward dilation multiplication. We’ll write image length as the scale factor times the unknown original length, place the given quantities in that equation, …
Preview problemFind the dilation scale factor from an original segment length and its image length
Verify that dilations scale line segments by the dilation scale factor.
The forward dilation factor compares each image length with its corresponding original length. We’ll keep those roles in order, form image divided by original, simplify the ratio rather than using …
Preview problemVerify that all sides of a polygon scale equally under dilation
Verify that dilations scale line segments by the dilation scale factor.
A dilation must use one multiplicative factor across every corresponding side, not one additive increase. We’ll align the side pairs in matching order, compute each image-to-original ratio independently, check that …
Preview problemUse a dilation scale factor to find a missing side in similar figures
Verify that dilations scale line segments by the dilation scale factor.
A missing corresponding side follows the same common scale factor as the rest of similar figures. We’ll confirm which value is the original and which position is the image, apply …
Preview problemDetermine whether transformation data could describe a dilation
Verify that dilations scale line segments by the dilation scale factor.
Similar size alone does not identify one dilation; the transformation also needs a common radial center. We’ll test all corresponding lengths for one positive image-to-original factor, check that every point-image …
Preview problemSeparate length, perimeter, and area dilation factors
Verify that dilations scale line segments by the dilation scale factor.
Dilation factors depend on dimensionality. We’ll scale generic side lengths by k, factor that same multiplier from the perimeter sum, multiply the two independently scaled dimensions for area, and use …
Preview problemDecide SSS similarity from three explicit ratios
Use similarity transformations to decide similarity and explain angle equality and side proportionality in triangles.
SSS similarity requires one consistent multiplicative relationship across all three corresponding sides. We’ll align shortest with shortest and continue in order, form every second-to-first ratio the same way, compare all …
Preview problemClassify AA similarity as proven, disproven, or insufficient
Use similarity transformations to decide similarity and explain angle equality and side proportionality in triangles.
AA asks whether two corresponding angle pairs are congruent, not whether side lengths are available or whether each triangle merely sums to a straight angle. We’ll match equal measures, count …
Preview problemClassify a transformation sequence by similarity invariants
Use similarity transformations to decide similarity and explain angle equality and side proportionality in triangles.
A transformation sequence is a similarity transformation when angles survive and every length receives one uniform composite factor. We’ll track those two invariants through the dilation, track them again through …
Preview problemDerive three separate vertex mappings in similar triangles
Use similarity transformations to decide similarity and explain angle equality and side proportionality in triangles.
A triangle similarity statement is an ordered record of three separate vertex correspondences. We’ll extract each mapping from the given congruent angles, place the target vertices in the same first-second-third …
Preview problemFind the scale factor between similar triangles from a pair of corresponding sides
Use similarity transformations to decide similarity and explain angle equality and side proportionality in triangles.
A similarity scale factor depends on both correspondence and transformation direction. We’ll decode the ordered triangle names to match the side pair, identify which triangle is the source and which …
Preview problemFind a missing side length from corresponding sides in similar triangles
Use similarity transformations to decide similarity and explain angle equality and side proportionality in triangles.
A missing side inherits the same factor as a fully known corresponding pair. We’ll establish the actual side matches from the vertex correspondence, compute the target-to-source factor from the known …
Preview problemFind a missing angle or variable from corresponding angles in similar triangles
Use similarity transformations to decide similarity and explain angle equality and side proportionality in triangles.
Corresponding angles in similar triangles are equal, but the written triangle order must identify the correct pair first. We’ll expand the first-second-third vertex mapping, locate the requested angle’s partner, transfer …
Preview problemFind an unknown measurement by comparing corresponding sides in similar triangles
Use similarity transformations to decide similarity and explain angle equality and side proportionality in triangles.
The shadow model works because both objects form right triangles under the same sunlight angle. We’ll justify AA, match vertical height with horizontal shadow in both triangles, keep both ratios …
Preview problemAudit and correct a similarity correspondence
Use similarity transformations to decide similarity and explain angle equality and side proportionality in triangles.
The ordered similarity statement is the authoritative correspondence record. We’ll expand it into three position-by-position vertex mappings, compare the disputed angle claim against the expected partner, identify the exact mismatch, …
Preview problemUse AA similarity to find a missing side
Use similarity transformations to establish the AA similarity criterion.
AA establishes that corresponding sides are proportional; it does not make their lengths equal. We’ll use the angle evidence to justify similarity, confirm the two stated side pairings, compute the …
Preview problemUse AA-similar triangles to match corresponding angles and find a missing angle or variable
Use similarity transformations to establish the AA similarity criterion.
The ordered similarity statement identifies corresponding angles before any calculation. We’ll expand the first-second-third vertex mapping, locate the requested angle’s partner, use congruence of corresponding angles to transfer the known …
Preview problemComplete an AA similarity proof
Use similarity transformations to establish the AA similarity criterion.
A complete AA proof needs two explicit angle pairs and an ordered correspondence. We’ll state both givens, record their vertex mappings, assign the only remaining vertex pair, place corresponding letters …
Preview problemDecide whether two triangles are similar by AA from angle evidence
Use similarity transformations to establish the AA similarity criterion.
AA is a sufficiency test with a precise evidence threshold. We’ll count only the angle relationships actually proved, compare that count with the two-pair requirement, distinguish insufficient evidence from evidence …
Preview problemWrite the correct triangle correspondence from matching angles for an AA similarity statement
Use similarity transformations to establish the AA similarity criterion.
A similarity statement must preserve every supplied vertex match position by position. We’ll list the three mappings, assemble the triangle names in the same order, derive each side correspondence by …
Preview problemMatch two angle pairs for AA similarity
Use similarity transformations to establish the AA similarity criterion.
Equal angle measures create the AA correspondence, not the triangles’ visual placement. We’ll match the two supplied measure pairs, apply AA, compute each remaining angle from the triangle sum as …
Preview problemUse parallel lines to establish AA similarity
Use similarity transformations to establish the AA similarity criterion.
Parallel lines create the needed angle relationships only through the relevant transversals. We’ll identify the shared vertex angle, use the parallel segments to justify a second corresponding pair, record the …
Preview problemCombine vertical and supplied angles for AA
Use similarity transformations to establish the AA similarity criterion.
This configuration combines one supplied angle pair with one theorem-generated pair. We’ll match the equal outer angles, identify the truly opposite angles at the intersection as vertical, use those two …
Preview problemEstablish AA similarity for right triangles
Use similarity transformations to establish the AA similarity criterion.
Right-triangle structure supplies one AA pair immediately, but one matching acute pair is still needed. We’ll pair the right angles, pair the supplied acute angles, invoke AA, compute the remaining …
Preview problemUse the side-splitter theorem to find a missing segment
Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.
The side-splitter theorem relates the two pieces on one triangle side to the corresponding two pieces on the other. We’ll verify the interior segment is parallel to the third side, …
Preview problemUse the converse of the side-splitter theorem to decide whether a segment is parallel to the third side of a triangle
Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.
The converse side-splitter theorem turns proportional division into a parallel-line conclusion. We’ll form the two split-side ratios in matching order, simplify them and confirm with cross products, verify the converse …
Preview problemUse the angle bisector theorem to find a missing segment
Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.
The angle-bisector theorem depends on pairing each adjacent side with the neighboring piece of the opposite side. We’ll confirm the bisector marks, build one consistently ordered proportion, solve it without …
Preview problemUse right-triangle similarity to find a missing length
Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.
An altitude to a right triangle’s hypotenuse creates a geometric-mean relationship, not an additive one. We’ll use the similar smaller triangles to place the altitude between the two hypotenuse pieces …
Preview problemUse the Pythagorean Theorem after a similarity proof
Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.
Once the triangle is known to be right, the key is assigning the two legs and the hypotenuse correctly. We’ll square and add the leg lengths, take the positive square …
Preview problemComplete a triangle-similarity theorem proof
Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.
The desired split-side ratio is hidden inside a whole-side similarity proportion. We’ll replace each whole side with the sum of its two pieces, clear the denominators, expand both products, cancel …
Preview problemChoose the similarity theorem needed for a diagram
Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.
The theorem’s direction is determined by what the problem gives and what it asks us to conclude. We’ll recognize the given parallel interior segment, identify the forward side-splitter theorem, compare …
Preview problemUse parallel segments to derive proportional sides
Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.
Parallelism connects two complementary views of this diagram: AA similarity and proportional side splitting. We’ll establish the correspondence, solve an upper-to-lower segment proportion, rebuild both whole sides, and verify that …
Preview problemDerive the Pythagorean Theorem from altitude similarity
Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.
Altitude similarity turns each leg into the geometric mean of the hypotenuse and that leg’s adjacent projection. We’ll derive one leg-square relation from each smaller triangle, keep the projection pairings …
Preview problemProve triangles congruent using a valid criterion
Use triangle congruence and similarity criteria to solve problems and prove geometric relationships.
A valid congruence criterion must match exactly the evidence supplied. We’ll inventory the three marked side pairs, note that no angle relationship is available, recover the vertex mapping from the …
Preview problemProve triangles similar using a valid criterion
Use triangle congruence and similarity criteria to solve problems and prove geometric relationships.
Two corresponding angle pairs already determine a triangle’s shape. We’ll match the marked pairs, apply AA, use the triangle angle sum to explain why the third pair must also agree, …
Preview problemDecide whether congruence or similarity is appropriate
Use triangle congruence and similarity criteria to solve problems and prove geometric relationships.
When every corresponding side is exactly equal, the evidence supports more than proportional shape. We’ll distinguish equality from a common ratio, apply the three-side congruence criterion, relate congruence to the …
Preview problem