NY Regents Geometry flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Rigid motion

A transformation that preserves distance and angle measure: translation, rotation, reflection, or any composition of these. Two figures are congruent exactly when some rigid motion maps one onto the other.

Triangle congruence: SAS

If two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another, the triangles are congruent. The angle must lie between the two named sides.

AA similarity criterion

If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. The third pair is then automatically congruent because triangle angles sum to 180 degrees.

Scale factor effects

For similar figures with linear scale factor k, corresponding lengths scale by k, areas scale by k squared, and volumes scale by k cubed. Use this to convert between length, area, and volume ratios.

Pythagorean theorem converse

If a triangle has sides a, b, and c with a squared plus b squared equal to c squared, then the angle opposite side c is a right angle. This classifies triangles as right, acute, or obtuse.

45-45-90 triangle ratios

An isosceles right triangle has legs of equal length x and hypotenuse x times the square root of 2. The two acute angles each measure 45 degrees.

30-60-90 triangle ratios

Sides are x, x times the square root of 3, and 2x, where x is opposite the 30-degree angle, x root 3 is opposite 60 degrees, and 2x is the hypotenuse.

Sine, cosine, tangent ratios

In a right triangle, sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent, all measured from a chosen acute angle.

Law of Cosines

For any triangle with sides a, b, c and angle C opposite side c: c squared equals a squared plus b squared minus 2ab cosine C. Use it with SAS or SSS information.

Inscribed angle theorem

An inscribed angle measures half its intercepted arc, while a central angle measures the full arc. Consequently, any angle inscribed in a semicircle is a right angle.

Cyclic quadrilateral angles

A quadrilateral whose four vertices lie on one circle is cyclic. Its opposite angles are supplementary, summing to 180 degrees, which allows finding unknown angles from given ones.

Tangent-radius perpendicularity

A tangent line meets a circle at exactly one point and is perpendicular to the radius drawn to that point. Also, two tangent segments from the same external point are congruent.

Intersecting chords theorem

When two chords intersect inside a circle, the product of the two segments of one chord equals the product of the two segments of the other chord.

Secant power theorem

From an external point, the tangent length squared equals the outside secant segment times the whole secant. For two secants, outside times whole is equal for both.

Arc length formula

Arc length equals the central angle divided by 360, multiplied by 2 pi r. The fraction of the circle's circumference matches the fraction of 360 degrees.

Sector area formula

Sector area equals the central angle divided by 360, multiplied by pi r squared. It is the same fraction of the circle's area as the arc is of the circumference.

Circle center-radius equation

The equation (x minus h) squared plus (y minus k) squared equals r squared describes a circle with center (h, k) and radius r. Expand and complete the square to recover it.

Distance formula

The distance between points (x1, y1) and (x2, y2) is the square root of the sum of the squared differences of the x-coordinates and y-coordinates.

Midpoint formula

The midpoint of a segment with endpoints (x1, y1) and (x2, y2) is the average of the x-coordinates paired with the average of the y-coordinates.

Perpendicular slope relationship

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, so the product of the slopes equals negative one. Vertical and horizontal lines are also perpendicular.

Parallel slope relationship

Two distinct nonvertical lines are parallel exactly when they have the same slope. Vertical lines are parallel to each other because their slopes are undefined.

Completing the square for circles

Rewrite x squared plus y squared plus Dx plus Ey plus F equals 0 by completing the square in x and y. The center is (-D/2, -E/2) and the radius follows from the constant.

Rectangle diagonal test

A parallelogram is a rectangle exactly when its diagonals are congruent. Using coordinates, compare the two diagonal lengths with the distance formula to confirm.

Rhombus diagonal test

A parallelogram is a rhombus exactly when its diagonals are perpendicular. Equivalently, all four sides are congruent, which can be checked with the distance formula.

Cylinder volume formula

The volume of a cylinder equals the base area pi r squared multiplied by the height h. Volume is expressed in cubic units, so all lengths must share the same unit.

Cone volume formula

The volume of a cone is one third of pi r squared times the height h. A cone holds one third the volume of the cylinder with the same base and height.

Sphere volume and surface area

A sphere of radius r has volume four thirds pi r cubed and surface area 4 pi r squared. Surface area is in square units and volume in cubic units.

Trapezoid area formula

The area of a trapezoid is one half the sum of the two parallel bases multiplied by the height, where height is the perpendicular distance between the bases.

Regular polygon area formula

Area equals one half the apothem times the perimeter. The apothem is the perpendicular distance from the center to a side, found using right-triangle trigonometry.

Cone lateral surface area

Lateral area of a cone equals pi times radius times slant height l, where l is the distance from apex to a base edge. Total surface area adds the base pi r squared.

Density relationship

Density equals mass divided by volume, so mass equals density times volume and volume equals mass divided by density. Units must be consistent before substituting values.

Angle of elevation and depression

The angle of elevation is measured upward from horizontal to a line of sight; the angle of depression is measured downward. They are equal when taken between the same two points.