Ohio OST 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. Translations, rotations, and reflections are rigid motions, so a figure and its image are always congruent.

Translation rule

A translation by vector (a, b) maps each point (x, y) to (x + a, y + b). Distances and angles are preserved, so the image is congruent to the preimage.

Rotation 90 counterclockwise

About the origin, a 90-degree counterclockwise rotation maps (x, y) to (-y, x). A 180-degree rotation maps (x, y) to (-x, -y).

Reflection across y = x

This reflection swaps coordinates, mapping (x, y) to (y, x). Reflection across the x-axis maps (x, y) to (x, -y), and across the y-axis to (-x, y).

Similarity transformation

A composition of rigid motions and dilations. It preserves angle measures and makes corresponding side lengths proportional, with the scale factor giving the ratio.

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 by the angle sum.

SSS and SAS similarity

SSS similarity requires all three pairs of corresponding sides proportional. SAS similarity requires two pairs of sides proportional with congruent included angles between them.

HL congruence

For right triangles, if the hypotenuse and one leg of one triangle are congruent to the hypotenuse and one leg of another, the triangles are congruent.

SSA is not a criterion

Two sides and a non-included angle do not guarantee congruence; the ambiguous case can produce zero, one, or two triangles. AAA guarantees similarity only, not congruence.

Triangle midsegment theorem

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and its length is half the length of that third side.

Triangle proportionality theorem

A line parallel to one side of a triangle that intersects the other two sides divides them into segments with proportional lengths. Its converse is also true.

Scale factor and area

If two similar figures have scale factor k, the ratio of corresponding lengths is k and the ratio of their areas is k squared. Perimeter also scales by k.

Coordinate proof setup

Place a figure with a vertex at the origin and sides along the axes, label coordinates with variables, then use distance, midpoint, and slope formulas to prove properties.

Parallelogram diagonal property

The diagonals of a parallelogram bisect each other. A quadrilateral is a parallelogram if its diagonals bisect each other, or if one pair of opposite sides is both parallel and congruent.

Rectangle and rhombus diagonals

A parallelogram is a rectangle exactly when its diagonals are congruent. It is a rhombus exactly when its diagonals are perpendicular and bisect opposite angles.

Circle standard equation

The circle with center (h, k) and radius r has equation (x - h)^2 + (y - k)^2 = r^2. Completing the square converts general form into this standard form.

Tangent-radius theorem

A tangent line is perpendicular to the radius drawn to the point of tangency. Two tangent segments drawn from the same external point are congruent.

Inscribed angle theorem

An inscribed angle measures half its intercepted arc, while a central angle equals its intercepted arc. An angle inscribed in a semicircle is a right angle.

Intersecting chords product

When two chords intersect inside a circle, the products of the two segments of each chord are equal: a times b equals c times d.

Arc length formula

Arc length equals (central angle in degrees divided by 360) times the circumference 2(pi)r. The same fraction of the circle gives sector area.

Sector area formula

Sector area equals (central angle in degrees divided by 360) times (pi)r squared. It is the same fraction of the full circle's area as the arc is of the circumference.

Cone volume formula

The volume of a cone is one third of (pi)r squared times height, or V = (1/3)(pi)r^2 h. It is one third the volume of a cylinder with the same base and height.

Sphere volume and surface area

A sphere has volume V = (4/3)(pi)r cubed and surface area S = 4(pi)r squared. Both depend only on the radius r.

Cylinder lateral area

The lateral surface area of a right cylinder is 2(pi)rh, where r is the base radius and h is the height. Total surface area adds the two circular bases.

Cone lateral area

The lateral surface area of a right cone is (pi)r times l, where l is the slant height. Slant height satisfies l squared = r squared + h squared.

Volume scaling rule

If a solid is scaled by factor k, its surface area scales by k squared and its volume scales by k cubed. Lengths scale by k.

Density formula

Density equals mass divided by volume. Population density equals number of people divided by area. Density is used to model real-world quantities per unit of measure.

Perpendicular slope relationship

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, meaning the product of the slopes equals negative one.

Distance formula

The distance between points (x1, y1) and (x2, y2) is the square root of ((x2 - x1)^2 + (y2 - y1)^2). It follows from the Pythagorean theorem.

Midpoint formula

The midpoint of a segment with endpoints (x1, y1) and (x2, y2) is ((x1 + x2)/2, (y1 + y2)/2), the average of the coordinates.

Special right triangles

In a 45-45-90 triangle, legs are equal and the hypotenuse is leg times the square root of 2. In a 30-60-90 triangle, sides are in ratio 1 to square root 3 to 2.

Geometric mean in right triangles

The altitude to the hypotenuse of a right triangle divides it into two similar triangles. The altitude is the geometric mean of the two hypotenuse segments.