Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
Page 10 of 22
Each problem type has four distinct practice variants. Open a preview to move among all four.
Identify two separate circle center-radius records
Prove that all circles are similar.
Each circle needs its own center-radius record before any similarity comparison is made. We’ll keep the named centers paired with their stated radii, then compare the radii multiplicatively while treating …
Preview problemFind the scale factor that maps one circle to another
Prove that all circles are similar.
A dilation scale factor is a directed multiplicative comparison, so mapping order determines the ratio. We’ll divide the target radius by the source radius and verify that multiplying the source …
Preview problemDescribe a dilation that maps one circle to another when the circles have the same center
Prove that all circles are similar.
When two circles share a center, that common point is the natural fixed point of the dilation. We’ll form the target-to-source radius ratio, check whether it enlarges or reduces, and …
Preview problemMap one circle by a specified translation-dilation composition
Prove that all circles are similar.
A translation-dilation composition handles circle location and size in separate ordered stages. We’ll subtract centers to align them first, compare radii for the scale factor, and center the dilation at …
Preview problemUse a circle similarity scale factor to compare circumferences
Prove that all circles are similar.
Circumference is a linear measure, so it scales like radius rather than area. We’ll substitute a scaled radius into the circumference formula, factor out the length scale, and preserve the …
Preview problemFind the area ratio of two similar circles from a scale factor
Prove that all circles are similar.
Area responds to a dilation in two dimensions, so its factor is the square of the linear scale. We’ll substitute the scaled radius into the circle-area formula, square the entire …
Preview problemComplete a missing step in a circle-similarity proof
Prove that all circles are similar.
A circle-similarity proof separates position from size: first align the centers, then scale every radius from that shared point. We’ll form the target-to-source radius ratio and verify that the resulting …
Preview problemVerify a coordinate similarity between circles
Prove that all circles are similar.
A coordinate similarity must map the center correctly and scale every radial displacement from that center by one constant. We’ll compute the center translation and radius ratio, track a boundary …
Preview problemFind an inscribed angle measure from its intercepted arc
Analyze relationships among inscribed angles, central angles, circumscribed angles, radii, chords, diameters, and tangents.
Before using an arc-angle rule, we need to identify where the angle’s vertex lies and which arc its sides intercept. We’ll confirm the angle is inscribed, match it to the …
Preview problemFind an intercepted arc measure from an inscribed angle
Analyze relationships among inscribed angles, central angles, circumscribed angles, radii, chords, diameters, and tangents.
When an inscribed angle is given and the intercepted arc is unknown, the familiar half-arc theorem must be reversed. We’ll identify the arc opposite the vertex, write the relationship in …
Preview problemFind a central angle measure from its intercepted arc
Analyze relationships among inscribed angles, central angles, circumscribed angles, radii, chords, diameters, and tangents.
A central angle is recognized by its vertex at the circle’s center and its sides along radii. We’ll match those radii to the intercepted arc and use their equal degree …
Preview problemCompare a central angle and an inscribed angle that intercept the same arc
Analyze relationships among inscribed angles, central angles, circumscribed angles, radii, chords, diameters, and tangents.
The central-versus-inscribed comparison is valid only after confirming both angles intercept the same pair of arc endpoints. We’ll express each angle in terms of that shared arc and eliminate the …
Preview problemUse a diameter to conclude an inscribed angle is a right angle
Analyze relationships among inscribed angles, central angles, circumscribed angles, radii, chords, diameters, and tangents.
A diameter determines a semicircle, but the right angle occurs at the third point on the circle rather than at a diameter endpoint. We’ll identify the inscribed angle intercepting that …
Preview problemFind the measure of an angle formed by two chords intersecting inside a circle from the intercepted arcs
Analyze relationships among inscribed angles, central angles, circumscribed angles, radii, chords, diameters, and tangents.
An angle formed by chords crossing inside a circle uses the two arcs intercepted by the angle and its vertical angle. We’ll source those opposite arcs, add their measures, and …
Preview problemFind the measure of an exterior angle formed outside a circle from two intercepted arcs
Analyze relationships among inscribed angles, central angles, circumscribed angles, radii, chords, diameters, and tangents.
An exterior secant angle is controlled by separation between two intercepted arcs, not their total. We’ll distinguish the far arc from the near arc, subtract in positive order, and take …
Preview problemUse the tangent-radius relationship to identify a perpendicular angle
Analyze relationships among inscribed angles, central angles, circumscribed angles, radii, chords, diameters, and tangents.
The tangent-radius theorem applies at the exact point where the radius meets the tangent line. We’ll identify the two rays meeting there, use perpendicularity, and name the right angle with …
Preview problemDerive a named chord, arc, or central-angle relation
Analyze relationships among inscribed angles, central angles, circumscribed angles, radii, chords, diameters, and tangents.
Congruent-circle-parts reasoning is a correspondence chain, so each chord must keep the same two endpoints as we move to arcs and central angles. We’ll confirm both chords lie in one …
Preview problemUse a perpendicular from the center to a chord to find the chord length
Analyze relationships among inscribed angles, central angles, circumscribed angles, radii, chords, diameters, and tangents.
The key geometric move happens before the Pythagorean theorem: a perpendicular from the center bisects the chord, so the unknown leg represents only half the chord. We’ll build the right …
Preview problemSolve for an unknown using equal tangent segments from the same external point
Analyze relationships among inscribed angles, central angles, circumscribed angles, radii, chords, diameters, and tangents.
Two tangent segments can be compared directly only when they start at the same external point and touch the same circle. We’ll verify that setup, set the segment lengths equal, …
Preview problemAnalyze the angle formed by two tangents drawn from the same external point
Analyze relationships among inscribed angles, central angles, circumscribed angles, radii, chords, diameters, and tangents.
For an exterior angle formed by two tangents, the relevant information is the gap between the intercepted arcs. We’ll order the major and minor arcs, take half their positive difference, …
Preview problemChoose and apply the correct circle-angle theorem from a mixed diagram description
Analyze relationships among inscribed angles, central angles, circumscribed angles, radii, chords, diameters, and tangents.
In a mixed circle diagram, the angle’s vertex location chooses the theorem before any arithmetic begins. We’ll identify the vertex on the circle, trace the rays to the intercepted arc’s …
Preview problemFind a missing angle in a cyclic quadrilateral using opposite supplementary angles
Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.
The word cyclic turns a four-angle problem into a two-angle pairing problem. We’ll identify opposite vertices rather than adjacent ones, then use their supplementary sum to recover the missing measure.
Preview problemDetermine whether a quadrilateral can be cyclic from its opposite angle pairs
Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.
Testing whether a quadrilateral can be cyclic uses the converse of the opposite-angle theorem. We’ll preserve the vertex order, form the two opposite pairs, and check each sum against a …
Preview problemSolve for a variable using supplementary opposite angles in a cyclic quadrilateral
Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.
The algebra comes after the geometry: first confirm that the variable angle and the given angle are at opposite vertices of a cyclic quadrilateral. We’ll set their expressions to a …
Preview problemComplete a missing step in an incenter, incircle, circumcenter, or circumcircle construction
Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.
A missing construction step depends on reading what has already been completed, not restarting the procedure. We’ll interpret what the intersecting perpendicular bisectors guarantee, determine what information that gives the …
Preview problemChoose a triangle center from the required equal distances
Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.
The decisive question is what must be equally distant: vertices or side lines. We’ll match the three corner-point distances to their defining loci, intersect two of those loci to identify …
Preview problemConstruct a circumcircle with perpendicular bisectors
Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.
A circumcircle starts with equal distances to the vertices, so the useful loci are side perpendicular bisectors rather than angle bisectors. We’ll construct two independent loci, justify their intersection through …
Preview problemFind and verify a coordinate circumcenter
Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.
Coordinate circumcenter work has two jobs: locate a candidate center and verify that it is equidistant from every vertex. We’ll exploit the horizontal and vertical sides to write simple perpendicular …
Preview problemConstruct an incircle with angle bisectors
Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.
An incircle center is chosen by equal distances to the side lines, so angle bisectors are the correct loci. We’ll intersect two of them, drop a perpendicular to any side …
Preview problemUse an incenter's perpendicular radius
Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.
For an incenter, the radius is not a segment to a vertex; it is a perpendicular distance to a side. We’ll read the named perpendicular foot, use that segment as …
Preview problemRelate inscribed angles to intercepted arcs
Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.
For an inscribed angle, the two chord endpoints determine the intercepted arc opposite the vertex. We’ll preserve that endpoint pairing, confirm that the vertex lies on the circle, and apply …
Preview problemIdentify the tangent points in the auxiliary-circle construction
Construct a tangent line from an external point to a circle.
The two-circle construction hides the tangent condition inside a diameter: each shared intersection subtends the auxiliary diameter as a right angle. We’ll identify those intersections, pair each with a radius …
Preview problemClassify a point relative to a circle before constructing tangents
Construct a tangent line from an external point to a circle.
Before constructing tangents, classify the point by comparing its center-to-point distance directly with the radius. We’ll use that comparison to determine the inside, on-circle, or outside case and then translate …
Preview problemSolve an unknown using equal tangent segments
Construct a tangent line from an external point to a circle.
Equal tangent lengths are a whole-segment relationship, provided both segments start at the same external point and touch the same circle. We’ll verify those conditions, set the complete segment labels …
Preview problemFind a tangent-triangle measure with the Pythagorean theorem
Construct a tangent line from an external point to a circle.
The tangent point creates the right angle, so the center-to-external-point segment—not the tangent segment—is the hypotenuse. We’ll assign the sides before writing the Pythagorean equation and then keep the nonnegative …
Preview problemComplete a missing step in the tangent-from-external-point construction
Construct a tangent line from an external point to a circle.
A missing-step question is about construction order: after the center-to-external-point segment is drawn, the next object must provide the center of the auxiliary circle that uses that segment as a …
Preview problem