KA Geometry Unit 1: Performing Transformations

What is the difference between a side, an angle, and a vertex in a polygon?

Sides are the line segments that form the polygon; angles are formed where the sides meet; vertices are the points where the sides meet.

What is a pentagon?

A polygon with five sides.

What is a hexagon?

A polygon with six sides.

What is a heptagon (or septagon)?

A polygon with seven sides.

What is an octagon?

A polygon with eight sides.

What is a nonagon?

A polygon with nine sides.

What is a decagon?

A polygon with ten sides.

What is a dodecagon?

A polygon with twelve sides.

How are polygons with many sides sometimes approximated?

Polygons with many sides (thousands or millions) are often approximated as circles.

What is a polyhedron?

A three-dimensional solid with flat polygonal faces, straight edges, and vertices.

What is a cube?

A polyhedron with six square faces.

What is a rectangular prism?

A polyhedron with six rectangular faces.

What is a tetrahedron?

A polyhedron with four triangular faces.

What is an octahedron?

A polyhedron with eight triangular faces.

What is a dodecahedron?

A polyhedron with twelve pentagonal faces.

What is an icosahedron?

A polyhedron with twenty triangular faces.

What is a sphere?

A three-dimensional object where all points on the surface are equidistant from the center.

What is a cylinder?

A three-dimensional figure with two parallel circular bases connected by a curved surface.

What is a cone?

A three-dimensional figure with a circular base that tapers to a point (apex).

What is a pyramid?

A polyhedron with a polygonal base and triangular faces that meet at a single point (apex).

What is a prism?

A polyhedron with two identical parallel polygonal bases and rectangular lateral faces.

What is a rigid transformation?

A transformation that changes the position of a set of points without changing their shape or size (distances between points remain the same).

What are the main types of rigid transformations?

Rotations, reflections, and translations.

What is dilation?

Scaling a shape up or down, changing its size but not its shape. It is NOT a rigid transformation.

What is a translation?

Moving every point of a figure the same distance in the same direction (sliding).

What properties are preserved under translation?

Length of line segments, angle measures, and parallelism of lines.

How do you represent a translation using coordinate notation?

(x, y) → (x + a, y + b), where 'a' is the horizontal shift and 'b' is the vertical shift.

How do you determine the vertical shift of a translation from point P to P'?

Subtract the y-coordinate of P from the y-coordinate of P' (y₂ - y₁).

If P is at (x, 3) and P' is at (x, 7), what is the vertical shift?

7 - 3 = 4 units upward.

If P is at (x, 7) and P' is at (x, 3), what is the vertical shift?

3 - 7 = -4 units downward.

What is a rotation?

Turning a figure around a fixed point (center of rotation).

What two parameters define a rotation?

The center of rotation and the angle of rotation.

What is a positive rotation?

Counterclockwise rotation.

What is a negative rotation?

Clockwise rotation.

How do you determine the direction of rotation (clockwise or counterclockwise)?

By tracing the shortest path from the original point to the rotated point. The shortest path will be the direction of rotation.

What is the rotation rule for 90 degrees counterclockwise?

(x, y) → (-y, x)

What is the rotation rule for 180 degrees counterclockwise?

(x, y) → (-x, -y)

What is the rotation rule for 270 degrees counterclockwise?

(x, y) → (y, -x)

What is the rotation rule for 360 degrees counterclockwise?

(x, y) → (x, y)

What is the rotation rule for 90 degrees clockwise?

(x, y) → (y, -x)

What is the rotation rule for 180 degrees clockwise?

(x, y) → (-x, -y)

What is the rotation rule for 270 degrees clockwise?

(x, y) → (-y, x)

What is the rotation rule for 360 degrees clockwise?

(x, y) → (x, y)

What is a reflection?

Mirroring a figure across a line (axis of reflection).

What happens to the coordinates when reflecting across the y-axis?

The x-coordinate changes sign, the y-coordinate stays the same. (x,y) -> (-x,y)

What happens to the coordinates when reflecting across the x-axis?

The y-coordinate changes sign, the x-coordinate stays the same. (x,y) -> (x,-y)

How do you find the line of reflection?

Find the midpoint between corresponding points of the original and reflected figures.

What is the midpoint formula?

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

What is dilation?

Scaling a figure up or down by a scale factor.

What is a scale factor?

a number that represents the ratio between the size of an original shape and its scaled copy

How do you find corresponding sides in dilated figures?

By matching the relative positions of the sides in the original and dilated figures.

How do you calculate the length of a dilated line segment?

To find the scale factor of a dilation, divide the length of a side in the dilated image by the length of the corresponding side in the original figure.

Length of dilated line segment/original image = scale factor

What is the formula for the length of a dilated shape?

Length of Line Segment * Scale Factor = Length of the Dilated Shape

What is the center of dilation?

The fixed point from which all points are scaled during dilation.

How do you find the center of dilation?

Draw lines connecting corresponding points of the original and dilated figures; the intersection of these lines is the center.

What happens when the scale factor is greater than 1?

The figure is enlarged (expanded).

What happens when the scale factor is less than 1 (but greater than 0)?

The figure is reduced (shrunk).

Who is considered the 'father of geometry'?

Euclid.

What is Euclid's famous work called?

The Elements.

When was The Elements written (approximately)?

300 BC.

What does 'geo' mean?

Earth.

What does 'metria' (or 'metery') mean?

Measure(ment).

What is the literal meaning of 'geometry'?

Measurement of the Earth.

What are dimensions?

Independent directions in space.

How many dimensions does a point have?

Zero.

How many dimensions does a line have?

One.

How many dimensions does a plane have?

Two.

What is a point?

A location in space with no size.

What is a line?

Extends infinitely in both directions.

What is a line segment?

Part of a line with two endpoints.

What is a plane?

A flat, two-dimensional surface.

What is the difference between a line and a line segment?

A line extends infinitely, a segment has endpoints.

What is a ray?

Has a starting point (vertex) and extends infinitely in one direction.

What are perpendicular lines?

Lines that intersect at a 90-degree angle.

What are parallel lines?

Lines in the same plane that never intersect.

What is an angle?

Two rays with a common vertex.

What is a vertex (plural: vertices)?

A point where lines, segments, or rays meet.

What is a circle?

Points equidistant from a center.

What is a chord?

A line segment connecting two points on a circle (not through the center).

What is a polygon?

A closed shape with straight sides.

What is a triangle?

A polygon with three sides.

What is an equilateral triangle?

All sides and angles are equal.

What is an isosceles triangle?

Two sides and two angles are equal.

What is a scalene triangle?

All sides and angles are different.

What is a right triangle?

One angle is 90 degrees.

What is an acute triangle?

All angles are less than 90 degrees.

What is an obtuse triangle?

One angle is greater than 90 degrees.

What is a quadrilateral?

A polygon with four sides.

What is a square?

All sides equal, all angles 90 degrees.

What is a rectangle?

Opposite sides equal, all angles 90 degrees.

What is a rhombus?

All sides equal, angles not necessarily 90 degrees.

What is a parallelogram?

Opposite sides equal and parallel.

What is a trapezoid (US) / trapezium (UK)?

One pair of parallel sides.

What is a kite?

Two pairs of adjacent equal sides.

How do you determine the angle of rotation?

By drawing a line from the center of rotation to the original point and the rotated point. The angle formed is the angle of rotation.

What if the direction of the rotation is 180 degrees or 360 degrees.

If it is 180, the point shifts (x,y) to (-x,-y) No matter whether it is clockwise or counterclockwise. Same thing for 360 degrees. (x,y) to (x,y) (NO CHANGE)

The rotation rule for (-) 90 degrees (clockwise) (x,y) becomes (y,-x) is equal to what rotation

(+) 270 degrees counterclockwise (x,y) becomes (y,-x)

The rotation rule for (+) 90 degrees (clockwise) (x,y) becomes (-y,x) is equal to what other rotation rule?

(-) 270 degrees counterclockwise (x,y) becomes (-y,x)

How are 90 and 270 degrees related?

They’re angle rotations that are equal to each other. 90 degrees clockwise and 270 degrees counterclockwise as well as 90 degrees counterclockwise and 270 degrees clockwise.

How are 90 and 270 degrees related? - In simple terms

The 90/270 rotation is equal to the inverse 270/90. So if you had +(90), the rotation is equal to -270.