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Geometry Module 1
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Gravity
Terms in this set (64)
Dilation
A transformation that changes the size of an object, but not the shape.
- A dilation DOES preserve the orientation of the figure.
- A dilation is NOT a rigid motion.
-Note that lines after a dilation map to parallel lines (unless the center of dilation is on the line, which maps to the same line).
What happens when a line is dilated and the center of dilation is on the line?
The image is the SAME LINE
What happens when a line is dilated and the center of dilation is NOT on the line?
The image is a PARALLEL line
Translation of a Figure (sliding motion)
Translation is a transformation that moves points the same distance and in the same direction.
-Move left or right FIRST, then up or down. (x then y)
- A Translation DOES preserve the orientation of the figure.
- A Translation is a rigid motion.
Example: (x-7, y-3) algebraic notation
Line reflection
A transformation that "flips" a figure over a mirror or reflection line.
- a line reflection DOES NOT preserve the orientation of the figure.
Reflection over x-axis (or any horizontal line)
-Count to the x-axis (or any horizontal line) and repeat on the other side
- A reflection over a line DOES NOT preserve the orientation of the figure.
- A reflection is a rigid motion.
Reflection over the y-axis (or any vertical line)
Count to the y-axis (or other vertical line) and repeat on the other side
- A reflection over a line DOES NOT preserve the orientation of the figure.
- A reflection is a rigid motion.
Reflection over the line y=x or y= -x
Count perpendicular to the line (diagonally) and repeat on the other side.
- A reflection over a line DOES NOT preserve the orientation of the figure.
- A reflection is a rigid motion.
Rotation 90 degrees clockwise
One quadrant clockwise.
- A rotation DOES preserve the orientation of the figure.
- A rotation is a rigid motion.
Rotation 90 degrees counterclockwise
One quadrant counterclockwise
- A rotation DOES preserve the orientation of the figure.
- A rotation is a rigid motion.
Midpoint formula
Count it on a graph (halfway between two points)
Distance Formula
Use distance formula or draw a right triangle, count the legs (a and b) and plug into Pythagorean Thm
Pythagorean Theorem
a²+b²=c²
Finding endpoint given one endpoint and Midpoint
Count on a graph (the slope) to the midpoint and repeat.
Simplify a Radical or "Simplest Radical Form"
Look for perfect squares (4,9,16,25,36...) inside the √. Factor them out and "unsquare" them.
ex. √48= √16 x √3 = 4√3
ex2. √72= √(36x2) = √36 x √2 = 6√2
Adding Radicals
Only like radicals can be added
Multiplying Radicals
multiply the coefficients, then multiply the numbers under the radical and then simplify that radical if necessary.
Rotation 180 degrees about the origin
Figure is upside down and 2 quadrants around the origin.
- A rotation DOES NOT change the orientation of the figure.
- A rotation is a rigid motion.
reflection through a point
Count to the point and repeat on the other side. ---Shape looks upside down after a reflection through a point.
-A reflection through a point DOES NOT change the orientation of the figure.
-Like rotating 180 degrees
Parallel Lines
lines in the same plane that never intersect
-Slopes are EQUAL
ex) 2/3 and 2/3
Perpendicular lines
Two lines that intersect to form right angles
-Slopes are OPPOSITE RECIPROCALS
ex) 3/4 and -4/3
Alternate Interior angles
Interior (interior) angles that lie on opposite sides of the transversal
- When lines are parallel, they are congruent.
-Theorem: When alternate interior angles are congruent, lines are parallel.
Alternate Exterior angles
Exterior (outside) angles that lie on opposite sides of a transversal.
- When lines are parallel, they are congruent.
-Theorem: When alternate exterior angles are congruent, lines are parallel.
Same Side Interior angles
The two angles that are inside (interior) the parallel lines and are on the same side of the transversal
-When lines are parallel they are SUPPLEMENTARY (add up to 180 degrees)
-Theorem: When same side interior angles are supplementary, lines are parallel.
Corresponding angles
lie on the same side of the transversal and in corresponding positions.
-When lines are parallel, they are congruent
-Theorem: When corresponding angles are congruent, lines are parallel.
Partitioning a Line Segment
Divide the slope (rise/run) by the total number of parts, graph additional points on the line.
Ex: Ratio is 3:2 ---> means 5 total parts. Divide rise/run by 5 and plot those points on the line, then go to the 3rd point.
Ex: Ratio is 2:5 ---> means 7 total parts. Divide rise/run by 7 and plot those points on the line, then go to the 2nd point.
Slope
Rise over run (Count from the left point, rise or fall first, then run to the right)
Area
the number of square units needed to cover a surface
Perimeter
the sum of the lengths of the sides of a polygon
Circumference of a circle
The distance around a circle.
If given the circumference, plug into 'C' and solve for 'r' using algebra to work backwards.
Area of a circle
A=πr²
The amount of space a circle takes up. If given the area, plug into 'A' and solve for 'r' using algebra to work backwards. (Divide by pi first and Square Root LAST)
Area of a Triangle
A=1/2
b
h or A=(bh)/2
Midpoint of a Line Segment
the point on a segment that divides it into two congruent segments
Bisector of a segment
line, ray segment, or plane that divides a segment into two congruent segments.
Perpendicular Bisector
A line that is perpendicular to a segment at its midpoint.
Angle Bisector
a ray that divides an angle into two congruent angles
Angle Bisector Theorem
If a point lies on the bisector of an angle, then it is equidistant from the two sides of the angle.
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Median of a Triangle
a segment from a vertex of a triangle to the midpoint of the opposite side
Altitude of a triangle (height)
The distance from a vertex of a triangle perpendicular to the opposite side.
Vertical Angles Theorem
-Vertical angles are opposite angles formed when two lines intersect.
-Vertical angles are always congruent
Complementary Angles
Angles whose sum is 90 degrees
Supplementary angles
Angles whose sum is 180 degrees
Construct congruent segments
Construct congruent angles
Construct an angle bisector.
Construct a perpendicular bisector
Construct a perpendicular line from a point not on the line
Construct a perpendicular line from a point on the line
Construct an equilateral triangle inscribed in a circle
Measure the radius first
-Connect every OTHER point ON the circle
Construct Hexagon inscribed in a circle
Measure a radius first
-Connect every point ON the circle
Construct a parallel line to a given line
Triangle Angle Sum Theorem
The sum of the measures of the angles of a triangle is 180.
The acute angles of a right triangle are
Complementary
Isosceles Triangle
A triangle that has 2 equal sides.
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Equilateral Triangle
A triangle with three congruent sides
Equilateral Triangle Theorem
If a triangle is equilateral, then it is equiangular
Area of Irregular Shape or Shaded Region
Find area of the simple shapes and either add them or subtract them, whichever is needed (or makes sense).
Exterior Angle of a Triangle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
Mapping A Figure Onto Itself
-a sequence of transformations where the image is in the same place as the pre-image after the transformation
-360 degrees divided by how many angles or vertices the regular figure has gives you the number of degrees to achieve rotational symmetry.
-This figure has 120 degree rotational symmetry, but it also has 240 degree rotational symmetry! (Think: multiple of 120)
Line Symmetry
a line that divides a figure into two halves that are mirror images of each other
HOY VUX
acronym for graphs of vertical and horizontal lines
scale factor of a dilation
the ratio of a side length of the image to the corresponding side length of the original figure
"Image DIVIDED by the PRE-IMAGE (or original)
-Image / Pre-Image
-2nd / 1st
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