Geometry Spring Final Exam Review Flashcards

Practice flashcards covering right triangles, similarity, polygons, circles, and volume/surface area concepts from the Geometry final exam review.

Pythagorean Theorem

A fundamental relation in geometry among the three sides of a right triangle stating that $a^2 + b^2 = c^2$.

Hypotenuse

The longest side of a right triangle, opposite the right angle.

Opposite Side

The side across from a given angle in a right triangle.

Adjacent Side

The side next to a given angle in a right triangle that is not the hypotenuse.

Soh

A trigonometric mnemonic representing $\text{Sin} = \frac{\text{Opposite}}{\text{Hypotenuse}}$.

Cah

A trigonometric mnemonic representing $\text{Cos} = \frac{\text{Adjacent}}{\text{Hypotenuse}}$.

Toa

A trigonometric mnemonic representing $\text{Tan} = \frac{\text{Opposite}}{\text{Adjacent}}$.

45-45-90 Triangle Ratios

The ratio of the sides in a triangle with angles $45^{\circ}$, $45^{\circ}$, and $90^{\circ}$, which is $1 : 1 : \sqrt{2}$.

30-60-90 Triangle Ratios

The ratio of the sides in a triangle with angles $30^{\circ}$, $60^{\circ}$, and $90^{\circ}$, which is $1 : \sqrt{3} : 2$.

Acute Triangle Classification

A triangle where the square of the longest side is less than the sum of the squares of the other two sides: $c^2 < a^2 + b^2$.

Obtuse Triangle Classification

A triangle where the square of the longest side is greater than the sum of the squares of the other two sides: $c^2 > a^2 + b^2$.

Right Triangle Classification

A triangle where the square of the longest side is exactly equal to the sum of the squares of the other two sides: $c^2 = a^2 + b^2$.

Similar Shapes Requirements

Two shapes are similar if their corresponding angles are congruent and their corresponding sides are proportional.

Scale Factor ($k$)

The ratio of the lengths of corresponding sides of an image to a pre-image, calculated as $k = \frac{\text{Image}}{\text{Pre-image}}$.

SSS Similarity

A theorem stating that if all three corresponding sides of two triangles are proportional, then the triangles are similar.

SAS Similarity

A theorem stating that if two pairs of corresponding sides are proportional and the included angles are congruent, then the triangles are similar.

AA Similarity

A postulate stating that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Perimeter Ratio

For similar shapes, the ratio of the sides is equal to the ratio of the perimeters.

Side-Splitter Theorem

If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those two sides proportionally.

Dilation

A transformation that produces an image that is the same shape as the original, but a different size (Enlargement or Reduction).

Reflection

A transformation that creates a mirror image of a figure across a specific line, such as the x-axis or y-axis.

Translation

A transformation that slides every point of a figure the same distance in the same direction.

Convex Polygon

A polygon in which all interior angles are less than $180^{\circ}$ and no diagonals lie outside the polygon.

Concave Polygon

A polygon that has at least one interior angle greater than $180^{\circ}$.

Regular Polygon

A polygon that is both equilateral (all sides congruent) and equiangular (all angles congruent).

Sum of Interior Angles

Calculated using the formula $(n - 2) \times 180^{\circ}$, where $n$ is the number of sides.

Sum of Exterior Angles

The sum of the measures of the exterior angles of any convex polygon is always $360^{\circ}$.

Polygon Diagonals Formula

The number of diagonals in a polygon with $n$ sides is calculated as $\frac{n(n - 3)}{2}$.

Midsegment (Median) of a Trapezoid

A segment connecting the midpoints of the legs of a trapezoid, with a length equal to $\frac{1}{2}(\text{base}_1 + \text{base}_2)$.

Parallelogram Properties

Opposite sides are parallel, opposite angles are congruent, and consecutive interior angles are supplementary.

Rectangle Diagonals

In a rectangle, the diagonals are congruent.

Rhombus Diagonals

In a rhombus, the diagonals are perpendicular and bisect the interior angles.

Square Diagonals

The diagonals of a square are both congruent and perpendicular.

Kite Diagonals

A kite has diagonals that are perpendicular to each other.

Isosceles Trapezoid Diagonals

A trapezoid where the legs are congruent and the diagonals are congruent.

Tangent Line

A line in the plane of a circle that intersects the circle in exactly one point and is perpendicular to the radius at that point.

Tangent Segments Congruence

Tangent segments drawn to a circle from the same external point are congruent.

Central Angle

An angle whose vertex is the center of the circle and whose measure is equal to the measure of its intercepted arc.

Inscribed Angle

An angle whose vertex is on the circle and whose measure is equal to $\frac{1}{2}$ the measure of its intercepted arc.

Inscribed Quadrilateral Theorem

Opposite angles of a quadrilateral inscribed in a circle are supplementary ($\text{Sum} = 180^{\circ}$).

Equation of a Circle

The standard form equation $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

Arc Length

A portion of the circumference of a circle, calculated based on the ratio of the central angle to $360^{\circ}$.

Area of a Circle

The measure of the interior surface of a circle, calculated as $A = \pi r^2$.

Circumference

The distance around a circle, calculated as $C = 2\pi r$ or $C = \pi d$.

Euler's Formula for Solids

A formula for polyhedrons stating that $\text{Faces} + \text{Vertices} = \text{Edges} + 2$.

Volume of a Prism

The space occupied by a prism, calculated as $V = Bh$, where $B$ is the area of the base.

Volume of a Pyramid

The space occupied by a pyramid, calculated as $V = \frac{1}{3}Bh$.

Volume of a Cone

The space occupied by a cone, calculated as $V = \frac{1}{3}\pi r^2 h$.

Volume of a Sphere

The space occupied by a sphere, calculated as $V = \frac{4}{3}\pi r^3$.

Surface Area of a Sphere

The total area of the outside of a sphere, calculated as $SA = 4\pi r^2$.