Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
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Each problem type has four distinct practice variants. Open a preview to move among all four.
Verify tangency from coordinate incidence and perpendicularity
Construct a tangent line from an external point to a circle.
Tangency in coordinates needs two independent facts: the contact point lies on both objects, and the candidate line is perpendicular to the radius there. We’ll verify incidence, compare directions, and …
Preview problemConstruct tangents with the midpoint auxiliary circle
Construct a tangent line from an external point to a circle.
The midpoint auxiliary circle is a device for manufacturing right angles, not just an extra circle. We’ll make the center-to-external-point segment a true diameter, use the two circles’ intersections to …
Preview problemVerify an external tangent with a semicircle right angle
Construct a tangent line from an external point to a circle.
The auxiliary diameter does two jobs: it proves a right angle at the circle intersection and turns the radius, center distance, and tangent segment into a right triangle. We’ll establish …
Preview problemAudit an external-tangent construction
Construct a tangent line from an external point to a circle.
Audit this construction by its invariant: the auxiliary circle must actually have the center-to-external-point segment as a diameter. We’ll test the proposed center and radius against endpoint incidence, correct the …
Preview problemConvert an angle from degrees to radians by multiplying by π/180
Derive arc length and sector area formulas using similarity; define radians and convert degrees/radians.
A unit-conversion factor should equal one while canceling the starting unit. We’ll orient the straight-angle equivalence with degrees in the denominator, simplify the exact pi fraction, and benchmark the result …
Preview problemConvert an angle measure from radians to degrees
Derive arc length and sector area formulas using similarity; define radians and convert degrees/radians.
To move from radians to degrees, orient the straight-angle equivalence so radians and the factor of pi cancel. We’ll simplify what remains in degrees and then check the result against …
Preview problemFind a central angle in radians from arc length and radius
Derive arc length and sector area formulas using similarity; define radians and convert degrees/radians.
Radians encode arc length as a multiple of the radius, so the central angle is the dimensionless arc-to-radius ratio. We’ll isolate the angle in the radian arc formula, cancel the …
Preview problemFind arc length from a radius and a central angle measured in radians
Derive arc length and sector area formulas using similarity; define radians and convert degrees/radians.
Since the angle is already in radians, arc length follows the direct scaling relationship without a degree conversion. We’ll keep pi exact, multiply the radius by the angular measure, and …
Preview problemFind arc length when the radius and central angle are given in degrees
Derive arc length and sector area formulas using similarity; define radians and convert degrees/radians.
Before using the compact radian arc formula, a degree measure must be converted; the same arc can also be viewed as a fraction of the circumference. We’ll use a unit-canceling …
Preview problemFind sector area when the central angle is given in radians
Derive arc length and sector area formulas using similarity; define radians and convert degrees/radians.
Sector area scales with the radius squared and with the same fraction of a full turn, which gives the radian formula one-half times radius squared times angle. We’ll verify the …
Preview problemFind sector area from a radius and a central angle in degrees by converting the angle to radians
Derive arc length and sector area formulas using similarity; define radians and convert degrees/radians.
The radian sector-area formula cannot take a degree measure directly, so the angle unit must be handled before any substitution. We’ll convert the central angle with a unit-canceling factor, preserve …
Preview problemFind a central angle in radians from arc length and radius
Derive arc length and sector area formulas using similarity; define radians and convert degrees/radians.
A radian angle is the ratio of intercepted arc length to radius, so the two length units should cancel. We’ll isolate the angle in the arc-length relationship, divide the sourced …
Preview problemFind the radius from an arc length and a central angle in radians
Derive arc length and sector area formulas using similarity; define radians and convert degrees/radians.
Because the angle is already in radians, the radius can be recovered directly from the arc-length relationship without a conversion. We’ll solve symbolically before substituting, divide arc length by angular …
Preview problemFind a missing sector measure from the sector area formula in radians
Derive arc length and sector area formulas using similarity; define radians and convert degrees/radians.
The unknown is an angle, so we’ll use the radian sector-area formula and isolate that factor rather than treating the drawing as measured. We’ll square the radius, simplify the coefficient …
Preview problemFind arc length from a radius and central angle by converting degrees to radians when needed
Derive arc length and sector area formulas using similarity; define radians and convert degrees/radians.
Circular travel is arc length: the rim point stays one radius from the center while sweeping through the given angle. Since that angle is already in radians, we’ll multiply radius …
Preview problemFind sector area from a radius and central angle by converting degrees to radians when needed
Derive arc length and sector area formulas using similarity; define radians and convert degrees/radians.
A sector covers the same fraction of a circle’s area as its central angle covers of a full turn. We’ll form that degree fraction, apply it to the full circle …
Preview problemFind a missing interior angle in a triangle using the 180° angle sum
Prove triangle theorems including angle sum, isosceles base angles, midsegment theorem, and medians concurrence.
The missing interior angle is the remainder after the two known angles account for part of a triangle’s fixed total. We’ll write all three angles in one sum, combine only …
Preview problemFind a triangle's exterior angle from the two remote interior angles
Prove triangle theorems including angle sum, isosceles base angles, midsegment theorem, and medians concurrence.
An exterior angle is tied to the two remote interior angles, not to the adjacent interior angle as an equal measure. We’ll identify the remote pair, add them by the …
Preview problemFind specified angles in an isosceles triangle
Prove triangle theorems including angle sum, isosceles base angles, midsegment theorem, and medians concurrence.
Equal sides determine equal opposite angles, so the first task is matching each marked side to the angle across from it. We’ll transfer the known base-angle measure to its partner …
Preview problemUse the converse of the isosceles triangle theorem to conclude two sides are equal
Prove triangle theorems including angle sum, isosceles base angles, midsegment theorem, and medians concurrence.
This is the converse direction of the isosceles theorem: equal angles tell us about their opposite sides. We’ll match each marked angle to the only side that does not touch …
Preview problemFind missing side lengths using the triangle midsegment theorem
Prove triangle theorems including angle sum, isosceles base angles, midsegment theorem, and medians concurrence.
The key is deciding which segment is the half-length: the midpoint connector is half the third side, not equal to it. We’ll verify both endpoint midpoint marks, pair the connector …
Preview problemClassify a segment using the median definition
Prove triangle theorems including angle sum, isosceles base angles, midsegment theorem, and medians concurrence.
A triangle segment is classified by where it begins and ends, not by how it looks. We’ll check that one endpoint is a vertex and the other is the midpoint …
Preview problemFind named centroid subsegments and the whole median
Prove triangle theorems including angle sum, isosceles base angles, midsegment theorem, and medians concurrence.
The centroid ratio is directional: the vertex-to-centroid piece has two parts while the centroid-to-midpoint piece has one. We’ll preserve the point order, use the known longer piece to find one …
Preview problemComplete a structured triangle-theorem reasoning chain
Prove triangle theorems including angle sum, isosceles base angles, midsegment theorem, and medians concurrence.
Two theorems work in sequence: the side marks first make the base angles equal, and the triangle total then determines their common measure. We’ll represent both with one variable, subtract …
Preview problemChoose the triangle theorem needed for a problem
Prove triangle theorems including angle sum, isosceles base angles, midsegment theorem, and medians concurrence.
The efficient way to identify a theorem is to match the direction from its evidence to its conclusion. We’ll start with the marked congruent sides, identify the opposite-angle relationship as …
Preview problemUse the opposite-sides theorem in a parallelogram to find a missing side measure or solve for a variable
Prove parallelogram theorems and their converses, including diagonal and rectangle results.
A parallelogram transfers a side length across the figure only to the side directly opposite it. We’ll trace the boundary order, identify the pair that shares no endpoint, write their …
Preview problemUse opposite angles in a parallelogram to find a missing angle or solve for a variable
Prove parallelogram theorems and their converses, including diagonal and rectangle results.
Opposite and adjacent angle pairs in a parallelogram use different relationships, so position must come before arithmetic. We’ll locate the requested vertex across from the given one, transfer the measure …
Preview problemUse supplementary consecutive angles in a parallelogram to find a missing angle measure
Prove parallelogram theorems and their converses, including diagonal and rectangle results.
Angles at consecutive vertices share a side, so they form the supplementary pair rather than the equal opposite pair. We’ll confirm that adjacency, write their sum as a straight-angle total, …
Preview problemUse the diagonal bisection property of a parallelogram to find a missing measure
Prove parallelogram theorems and their converses, including diagonal and rectangle results.
Diagonal bisection pairs the two pieces of the same diagonal, not pieces taken from different diagonals. We’ll preserve the point order through the intersection, equate the two halves on that …
Preview problemTest the opposite-sides converse for a parallelogram
Prove parallelogram theorems and their converses, including diagonal and rectangle results.
This is a converse test, so every part of the theorem’s hypothesis must be established before classifying the quadrilateral. We’ll list the boundary sides in order, verify two distinct opposite …
Preview problemTest the opposite-angles converse for a parallelogram
Prove parallelogram theorems and their converses, including diagonal and rectangle results.
A converse test requires the full hypothesis, so one matching angle pair cannot settle the classification. We’ll use cyclic vertex order to pair first with third and second with fourth, …
Preview problemTest the diagonal-bisection converse for a parallelogram
Prove parallelogram theorems and their converses, including diagonal and rectangle results.
The diagonal converse needs mutual bisection: the same intersection must be the midpoint of both diagonals. We’ll trace each diagonal through the shared point, verify its own two halves are …
Preview problemUse congruent diagonals in a rectangle to find a missing measure
Prove parallelogram theorems and their converses, including diagonal and rectangle results.
The rectangle theorem compares the two full diagonals, so the first job is distinguishing them from sides and half-diagonal pieces. We’ll connect each pair of opposite vertices, write the whole-to-whole …
Preview problemChoose the sufficient condition that proves a quadrilateral is a parallelogram
Prove parallelogram theorems and their converses, including diagonal and rectangle results.
To identify a sufficient-condition theorem, match both the type of evidence and the direction of the implication. We’ll verify that the marks cover two complete opposite-side pairs, require a conclusion …
Preview problemComplete a parallelogram proof by choosing the missing converse theorem
Prove parallelogram theorems and their converses, including diagonal and rectangle results.
A proof cannot use a property of an already-known parallelogram when the parallelogram classification is what must be proved. We’ll translate both given equalities into opposite-side congruences, check that the …
Preview problemUse vertical angles to find a missing angle measure or variable
Prove theorems about lines and angles: vertical angles, parallel-line angle relationships, and perpendicular bisectors.
At an intersection, position determines whether to copy a measure or take its supplement. We’ll confirm that the requested region is directly opposite the given angle, apply vertical-angle congruence, and …
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