Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
Page 12 of 22
Each problem type has four distinct practice variants. Open a preview to move among all four.
Use corresponding angles with parallel lines
Prove theorems about lines and angles: vertical angles, parallel-line angle relationships, and perpendicular bisectors.
Corresponding angles repeat the same corner position at the two intersections, but their congruence depends on the lines being parallel. We’ll verify the parallel marks and transversal, compare the relative …
Preview problemUse alternate interior angles with parallel lines
Prove theorems about lines and angles: vertical angles, parallel-line angle relationships, and perpendicular bisectors.
An alternate-interior pair must satisfy two location tests: both angles lie between the parallel lines, and they sit on opposite sides of the transversal. We’ll verify those positions first, invoke …
Preview problemUse alternate exterior angles with parallel lines
Prove theorems about lines and angles: vertical angles, parallel-line angle relationships, and perpendicular bisectors.
For alternate-exterior angles, both regions must be outside the parallel lines and on opposite sides of the transversal. We’ll confirm those two position tests, use the congruent-angle relationship, and check …
Preview problemUse same-side interior angles with parallel lines
Prove theorems about lines and angles: vertical angles, parallel-line angle relationships, and perpendicular bisectors.
Same-side interior angles lie between the parallel lines on one side of the transversal, so they are supplementary rather than congruent. We’ll verify that placement, write their sum as a …
Preview problemSelect a parallel-line converse from angle evidence
Prove theorems about lines and angles: vertical angles, parallel-line angle relationships, and perpendicular bisectors.
This argument starts with an angle relationship and seeks a line relationship, so the theorem direction must run from congruence to parallelism. We’ll classify the marked pair by its repeated …
Preview problemConclude two distances are equal from a point on a perpendicular bisector
Prove theorems about lines and angles: vertical angles, parallel-line angle relationships, and perpendicular bisectors.
A point on a perpendicular bisector is linked to the two endpoints of the original segment, not to the midpoint or the bisector line itself. We’ll verify the midpoint and …
Preview problemUse the converse of the perpendicular bisector theorem
Prove theorems about lines and angles: vertical angles, parallel-line angle relationships, and perpendicular bisectors.
Here the equal-distance fact is the input, so we need the converse that turns distances into a location. We’ll identify the common point and the two other endpoints, determine which …
Preview problemTest both perpendicular-bisector conditions
Prove theorems about lines and angles: vertical angles, parallel-line angle relationships, and perpendicular bisectors.
The name perpendicular bisector contains two independent tests, and neither one alone is enough. We’ll check that the candidate line passes through the segment’s midpoint, separately check that it forms …
Preview problemCombine line, angle, and perpendicular-bisector facts in a multi-step problem
Prove theorems about lines and angles: vertical angles, parallel-line angle relationships, and perpendicular bisectors.
At an intersection, the directly opposite region uses equality while an adjacent region uses a supplementary relationship. We’ll locate the requested angle relative to the given one, classify the pair …
Preview problemComplete a proof involving lines, angles, or perpendicular bisectors
Prove theorems about lines and angles: vertical angles, parallel-line angle relationships, and perpendicular bisectors.
The proof begins with angle congruence and must establish a line relationship, so a forward theorem would run in the wrong direction. We’ll verify that the marked angles are interior …
Preview problemChoose the correct theorem or converse for lines, angles, and perpendicular bisectors
Prove theorems about lines and angles: vertical angles, parallel-line angle relationships, and perpendicular bisectors.
The quickest theorem check is to separate what is already known from what must be concluded. We’ll use the given parallelism to establish the forward direction, classify the two marked …
Preview problemUse Cavalieri's principle to compare volumes
Explain circumference, area, and volume formulas using informal arguments such as dissection, Cavalieri, and limits.
Cavalieri’s principle compares solids layer by layer, so matching overall shapes is not the relevant test. We’ll verify the common height, compare cross-sectional areas at every corresponding level, imagine equal-thickness …
Preview problemChoose the informal argument that supports a geometric formula
Explain circumference, area, and volume formulas using informal arguments such as dissection, Cavalieri, and limits.
An area argument should preserve the region while converting it into a shape with a familiar formula. We’ll distinguish perpendicular height from the slanted side, identify a movable overhang, track …
Preview problemConnect the two circumference formulas
Explain circumference, area, and volume formulas using informal arguments such as dissection, Cavalieri, and limits.
The two circumference forms describe the same circle using different size measures, so the bridge between them is the diameter-radius relationship. We’ll replace diameter with two radii, simplify each expression …
Preview problemRecover circle area by rearranging sectors
Explain circumference, area, and volume formulas using informal arguments such as dissection, Cavalieri, and limits.
The rearrangement is useful because it trades a curved region for a near-parallelogram without changing area. We’ll track which arcs form one long base, relate that base to half the …
Preview problemBuild prism and cylinder volume from congruent layers
Explain circumference, area, and volume formulas using informal arguments such as dissection, Cavalieri, and limits.
A prism’s volume can be organized as a constant cross-sectional area repeated through a perpendicular distance. We’ll identify the area shared by every parallel layer, distinguish the height from any …
Preview problemRecover pyramid volume from a matched prism
Explain circumference, area, and volume formulas using informal arguments such as dissection, Cavalieri, and limits.
The matched prism provides an easier whole whose volume comes from base area and perpendicular height. We’ll find that whole first, use the stated decomposition to determine one equal share, …
Preview problemRecover cone volume from a matched cylinder
Explain circumference, area, and volume formulas using informal arguments such as dissection, Cavalieri, and limits.
The cone is easiest to measure through its matched cylinder, but the comparison works only when their bases and perpendicular heights agree. We’ll build the shared circular base area, find …
Preview problemBuild volume by accumulating cross-sections
Explain circumference, area, and volume formulas using informal arguments such as dissection, Cavalieri, and limits.
A slice area alone is not volume; each layer must also carry its thickness. We’ll form one thin-layer contribution, sum those contributions across the full height, use the supplied square-sum …
Preview problemFind the volume of a cylinder
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Cylinder volume is circular base area extended through a perpendicular height. We’ll map the labeled radius and height to their distinct roles, square only the radius when forming the base …
Preview problemFind the volume of a cone
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
A cone uses the same circular base and perpendicular-height product as a matched cylinder, together with its volume fraction. We’ll identify the radius and true height, form the cylinder-sized product …
Preview problemFind the volume of a pyramid
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
The base measurement is already an area, so it should enter the pyramid formula directly rather than be squared or rebuilt from a side length. We’ll pair that area with …
Preview problemFind the volume of a sphere
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
A sphere has one size input, so the first check is whether the diagram gives a radius or a diameter. We’ll map the center-to-surface length to the sphere formula, cube …
Preview problemFind a solid's height from volume and base measure
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Because the unknown is a dimension rather than the volume, the cylinder relationship must be rearranged around the height. We’ll form the circular base area from the radius, divide the …
Preview problemFind a missing radius from the volume of a cylinder, cone, or sphere
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
The unknown radius is buried inside the circular base area, so solving for it requires two stages. We’ll substitute the known volume and height, divide away every factor outside the …
Preview problemCompare two exact solid volumes by relation, ratio, and difference
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Because the solids share a radius and height, their common base-times-height factor makes the formula difference especially visible. We’ll write both exact volumes, keep pi symbolic, preserve the requested A-to-B …
Preview problemSolve a composite-volume problem
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
The first decision in a composite solid is whether a piece is attached or removed. We’ll represent the cavity with subtraction, calculate the full cylinder and cone separately using their …
Preview problemSolve a real-world capacity problem using volume
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Capacity is a volume question, so the container’s interior shape determines the model. We’ll treat the tank as a cylinder, build its circular base area from the radius, extend that …
Preview problemUse volume with unit conversion
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
A linear conversion factor acts once in each independent direction, so a volume conversion must use its third power. We’ll find the box’s volume in the given units, cube the …
Preview problemChoose a volume formula for a described solid
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Formula identification begins with the solid’s structure, not with matching familiar symbols. We’ll classify the can from its congruent circular bases and lack of taper, write the area of one …
Preview problemFind the new length after a dilation by scale factor k
Apply scale factors to length, area, and volume using k, k^2, and k^3 relationships.
A scale factor is a multiplicative comparison, and a single length uses that factor to the first power. We’ll identify the original measure and the forward dilation factor, multiply rather …
Preview problemFind the new area after a dilation
Apply scale factors to length, area, and volume using k, k^2, and k^3 relationships.
Area contains two independent length directions, so a dilation applies the linear factor twice. We’ll convert the stated scale factor into an area multiplier, apply that multiplier to the original …
Preview problemFind the new volume after a dilation
Apply scale factors to length, area, and volume using k, k^2, and k^3 relationships.
Volume tracks three independent length directions, so the dilation factor appears three times in the multiplier. We’ll cube the linear factor before touching the original volume, apply that derived multiplier …
Preview problemFind the scale factor from one length to another
Apply scale factors to length, area, and volume using k, k^2, and k^3 relationships.
The direction of the change controls the scale-factor ratio. We’ll label the starting length as the original and the ending length as the image, divide image by original instead of …
Preview problemFind the linear scale factor from an area ratio
Apply scale factors to length, area, and volume using k, k^2, and k^3 relationships.
The given comparison is two-dimensional, but the requested factor is linear, so the area relationship must be reversed. We’ll preserve the image-to-original ratio, set it equal to the square of …
Preview problemFind the linear scale factor from a volume ratio of similar solids
Apply scale factors to length, area, and volume using k, k^2, and k^3 relationships.
A volume ratio contains three copies of the linear scale factor, so it cannot be used directly as a length multiplier. We’ll keep the image-to-original direction, relate the ratio to …
Preview problem