80 Questions
What is the primary characteristic that distinguishes rigid transformations from non-rigid transformations?
The size of the preimage is not changed.
What is the term for a transformation that changes the location of the preimage but not its size, shape, or orientation?
Translation
What is the key rule that applies to all types of rigid transformations?
The angles being transformed always remain constant.
Which of the following is an example of a non-rigid transformation?
Shear
What is the term for a transformation that involves rotating the preimage around a fixed point?
Rotation
What is the term used to describe the original shape or figure before a transformation is applied?
Preimage
What does it mean when two shapes are described as similar?
They have the same shape but not the same size.
What is true about the perimeter of similar triangles?
It is always proportional to the ratio of side lengths.
What is an altitude in a triangle?
A line segment from a vertex that intersects the line containing the opposite side at a right angle.
What is a necessary condition for two polygons to be similar?
They must have the same shape and proportional sides.
What is true about corresponding angles of similar triangles?
They are always equal.
What is a median in a triangle?
A line segment extending from a vertex to the midpoint of the opposite side.
What is the main characteristic of a reflection in geometry?
The shape and orientation of the preimage are not changed.
When the line of reflection is the x-axis, how does the point (x, y) transform?
It transforms into (x, -y).
What is the effect of a scale factor of 2 in a dilation?
The image will stretch to twice its size.
What is the result of a 90-degree counterclockwise rotation around the origin?
The point (x, y) transforms into (-y, x).
What is the center of dilation in a dilation transformation?
A fixed point that does not move during the transformation.
What is the result of a reflection around the point of origin?
The point (x, y) transforms into (-x, -y).
What is a key characteristic of a tessellation?
It is a space-filling arrangement of plane figures that do not overlap or leave gaps.
What is the sum of the interior angles of every triangle?
180 degrees
What type of triangle has three acute angles?
Equilateral triangle
What is the purpose of the Pythagorean Theorem?
To find the sides of a right triangle
What is the term for a triangle with two sides of equal length?
Isosceles triangle
What is a characteristic of a regular tessellation?
It consists of only one repeated polygon.
What is the Pythagorean theorem used for?
In many real-life situations, including construction, physics, architecture, and landscaping
What is the formula to find the length of the hypotenuse using the Pythagorean theorem?
Square the length of one leg, square the length of the other leg, add these two results, and take the square root of the sum
What is an angle in geometry?
Two rays drawn in different directions from a shared endpoint
What is a linear pair in geometry?
A pair of angles that share a common vertex and a common side, and the two non-common sides create a straight line
What is a polyhedron?
A three-dimensional solid with faces that are all flat
What is Euler's characteristic?
An equation that governs the number of faces, vertices, and edges in a convex polyhedron
What is the dimension of a cone that is described as the distance from the perimeter of the base to the apex?
Slant height
What is the mathematical constant equal to 3.14 used in the equation to find the area of the base of a cone?
π
What is the type of probability that is the probability of two or more events?
Sequential probability
What is the scale on which probability is measured?
0 to 1
What is the term used to describe the likelihood that an event will happen?
Probability
What is the formula used to find the area of the base of a cone?
A = πr^2
What is the type of probability that is used to estimate what should occur with certain events?
Theoretical probability
What is the name of the three-dimensional object with a circular base that narrows as it approaches its tip or apex?
Cone
What is the term used to describe the probability of one event?
Simple probability
What is the name of the object that appears to have had its tip cut off?
Frustum
What is the result of a 270-degree counterclockwise rotation around the origin?
(x, y) ---> (-y, x)
What happens when a figure is dilated with a scale factor of 0.5?
The image will shrink towards the center of dilation.
What is the definition of a line of reflection?
A line perpendicular to the preimage and image.
What is the result of a reflection around the point of origin?
(x, y) ---> (-x, -y)
What is the difference between a translation and a dilation?
Translations maintain the same size and shape, while dilations do not.
What is the effect of a -90 degree rotation around the origin?
(x, y) ---> (-y, x)
What is the purpose of the Pythagorean theorem?
To find the length of the hypotenuse of a right triangle
What is a characteristic of a convex polyhedron?
It has no vertices or edges that go 'into' the polyhedron
What is the term for a pair of angles that add up to 90 degrees?
Complementary angles
What is Euler's characteristic used to describe?
The number of faces, edges, and vertices in a convex polyhedron
What is the term for a three-dimensional solid with flat faces?
Polyhedron
What is the term for a angle that is created from two rays drawn in different directions from a shared endpoint?
Angle
What is the main difference between a preimage and an image in a geometric transformation?
The preimage is the original shape, while the image is the transformed shape.
Which type of transformation does not change the size or shape of the preimage?
Rotation
What is a characteristic of a translation in geometry?
It changes the location of the preimage.
What is the term for a transformation that changes the size of the preimage but not its shape?
Dilation
What is true about the angles of a shape after a rigid transformation?
They remain constant.
What is a common term used to describe a rotation in geometry?
Turn
What is the main characteristic of a semi-regular tessellation?
A tessellation consisting of two or more types of repeated regular polygons.
What is the sum of the interior angles of every triangle?
180 degrees
What type of triangle has all three sides of equal length?
Equilateral triangle
What is the purpose of the Pythagorean theorem?
To find the length of a side of a triangle given the lengths of the other two sides.
What is a key characteristic of a regular tessellation?
It consists of only one repeated polygon.
What is the name of the equation used to find the length of the hypotenuse of a right triangle?
The Pythagorean theorem
What is the primary characteristic of congruent shapes?
They have the same shape and the same size.
What is true about the lengths of special segments in similar triangles?
They are proportionate to the ratio of side lengths.
What is the purpose of determining if two shapes are similar?
To find the lengths of missing sides or the degrees of unknown angles.
What is a characteristic of similar figures?
They have the same shape, angle sizes, and ratios of side lengths.
What is true about the corresponding angles of similar triangles?
They are always equal in size.
What is a way to determine if two polygons are similar?
By checking if they look the same, have the same angle sizes, and have proportional sides.
What is the dimension of a cone that is described as the distance from a point that is at a right angle to the base?
Height
What is the mathematical constant used in the equation to find the area of the base of a cone?
3.14
What type of probability is used to estimate what should occur with certain events?
Theoretical probability
What is the name of the three-dimensional object with a circular base that narrows as it approaches its tip or apex?
Cone
What is the term used to describe the likelihood that an event will happen?
Probability
What is the scale on which probability is measured?
0 to 1
What is the type of probability that is the probability of two or more events?
Sequential probability
What is the formula used to find the area of the base of a cone?
πr^2
What is the term used to describe the object that appears to have had its tip cut off?
Frustum
What is the type of probability that is the probability of one event?
Simple probability
Study Notes
Transformations in Geometry
- A transformation in math is a mapping of a preimage to an image of the same shape or function.
- There are two categories of transformations: rigid and non-rigid transformations.
- Rigid transformations:
- Do not change the size or shape of the preimage.
- Include translations, rotations, and reflections.
- Non-rigid transformations:
- Can change the size and shape of the preimage.
- Include shears and dilations.
Types of Transformations
- Translation:
- A rigid transformation that changes the location of the preimage.
- Also called a slide.
- Rotation:
- A rigid transformation that rotates the preimage around a fixed point.
- Also called a turn.
- Reflection:
- A rigid transformation that flips the preimage across a line.
- Also called a flip.
- Dilation:
- A non-rigid transformation that changes the size of the preimage.
- Also called an expansion or compression.
Reflections
- A reflection is a type of geometric transformation that flips an object across a line.
- The line of reflection is perpendicular to the preimage and image.
- To find the reflection of an object:
- Draw a coordinate plane.
- Extend a line segment from each point in the preimage to the line of reflection.
- Extend the line in the same direction by the same distance.
Rotations
- A rotation is a rigid transformation that turns an object around a fixed point.
- There are clockwise and counterclockwise rotations of 90, 180, and 270 degrees around the origin.
- Rotation rules:
- 90 degree rotation: (x, y) → (-y, x)
- 180 degree rotation: (x, y) → (-x, -y)
- 270 degree rotation: (x, y) → (y, -x)
Dilations
- A dilation is a transformation that changes the size of a figure using a center of dilation.
- There are three ways a scale factor can affect a preimage:
- Multiply by a value greater than 1 to stretch away from the center.
- Multiply by a value less than 1 to shrink towards the center.
- Multiply by a value equal to 1 to stay the same size.
Similarity and Congruence
- Similar shapes have the same shape and proportional corresponding sides.
- Congruent shapes have the same shape and size.
- Similar triangles have proportional corresponding sides and equal corresponding angles.
- Congruent triangles have equal corresponding sides and equal corresponding angles.
Triangles
- A triangle is a shape with three vertices, three sides, and three interior angles.
- Types of triangles:
- Scalene triangles: no sides of equal length.
- Isosceles triangles: two sides of equal length.
- Equilateral triangles: all sides of equal length.
- Acute triangles: all angles acute (less than 90 degrees).
- Obtuse triangles: one angle obtuse (greater than 90 degrees).
- Right triangles: one right angle (90 degrees).
Tessellations
- A tessellation is a repeating pattern of shapes that fit together without overlapping.
- Regular tessellations consist of one repeated polygon.
- Semi-regular tessellations consist of two or more types of repeated polygons.
Pythagorean Theorem
- The Pythagorean theorem states that a^2 + b^2 = c^2 in a right triangle.
- The theorem is useful for finding the length of the hypotenuse or a leg in a right triangle.
Angle Relationships
- Angles are formed by two rays drawn from a shared endpoint.
- Angle relationships:
- Complementary angles: add to 90 degrees.
- Supplementary angles: add to 180 degrees.
- Adjacent angles: share a common vertex and side.
- Linear pair: two angles that add to 180 degrees and share a common vertex and side.
- Vertical angles: two angles formed by intersecting lines.
Three-Dimensional Shapes
- A polyhedron is a three-dimensional solid with flat faces.
- A regular polyhedron (Platonic solid) has all faces that are regular polygons that are congruent to each other.
- Convex polyhedra have no vertices or edges that go "into" the polyhedron.
Cones
- A cone is a three-dimensional object with a circular base that narrows to a tip or apex.
- Dimensions:
- Height (h): distance from apex to a point at a right angle to the base.
- Radius (r): distance from center of base to perimeter.
- Slant height (s): distance from perimeter of base to apex.
- Area of base (b): area of circular base in square units.
Probability
- Probability is the likelihood that an event will happen, ranging from 0 to 1.
- Types of probability:
- Simple probability: probability of one event.
- Sequential probability: probability of two or more events.
- Events can be dependent or independent.### Probability in Real-Life Situations
- Probability is widely used in everyday life to predict outcomes, such as weather forecasts and game-winning chances.
Theoretical Probability
- Theoretical probability is a method to express the likelihood of an event occurring.
- It is calculated by dividing the number of favorable outcomes by the total possible outcomes.
- The result is a ratio that can be expressed as a fraction and a probability value between 0 and 1.
- The probability value can be easily converted into a percentage.
When to Use Theoretical Probability
- Theoretical probability is appropriate when:
- The researcher has intimate knowledge of the subject.
- The researcher can determine all possible and all favorable outcomes.
- Direct experimentation is not possible.
Theoretical vs. Experimental Probability
- Theoretical probability relies on logic and thorough knowledge of the subject.
- Experimental probability uses tests and experiments to determine probability.
- Both are branches of mathematics called probability, which addresses the likelihood of an event occurring under certain conditions.
Measuring Probability
- The scale of measuring probability ranges from 0 to 1.
- 0 represents an impossible event.
- 0.25 represents an unlikely event.
- 0.5 represents an even chance of an event occurring.
- 0.75 represents a likely event.
- 1 represents a certain event.
Experimental Probability Formula
- The formula for calculating experimental probability after trials is: [insert formula].
Transformations in Geometry
- A transformation in math is a mapping of a preimage to an image of the same shape or function.
- There are two categories of transformations: rigid and non-rigid transformations.
- Rigid transformations:
- Do not change the size or shape of the preimage.
- Include translations, rotations, and reflections.
- Non-rigid transformations:
- Can change the size and shape of the preimage.
- Include shears and dilations.
Types of Transformations
- Translation:
- A rigid transformation that changes the location of the preimage.
- Also called a slide.
- Rotation:
- A rigid transformation that rotates the preimage around a fixed point.
- Also called a turn.
- Reflection:
- A rigid transformation that flips the preimage across a line.
- Also called a flip.
- Dilation:
- A non-rigid transformation that changes the size of the preimage.
- Also called an expansion or compression.
Reflections
- A reflection is a type of geometric transformation that flips an object across a line.
- The line of reflection is perpendicular to the preimage and image.
- To find the reflection of an object:
- Draw a coordinate plane.
- Extend a line segment from each point in the preimage to the line of reflection.
- Extend the line in the same direction by the same distance.
Rotations
- A rotation is a rigid transformation that turns an object around a fixed point.
- There are clockwise and counterclockwise rotations of 90, 180, and 270 degrees around the origin.
- Rotation rules:
- 90 degree rotation: (x, y) → (-y, x)
- 180 degree rotation: (x, y) → (-x, -y)
- 270 degree rotation: (x, y) → (y, -x)
Dilations
- A dilation is a transformation that changes the size of a figure using a center of dilation.
- There are three ways a scale factor can affect a preimage:
- Multiply by a value greater than 1 to stretch away from the center.
- Multiply by a value less than 1 to shrink towards the center.
- Multiply by a value equal to 1 to stay the same size.
Similarity and Congruence
- Similar shapes have the same shape and proportional corresponding sides.
- Congruent shapes have the same shape and size.
- Similar triangles have proportional corresponding sides and equal corresponding angles.
- Congruent triangles have equal corresponding sides and equal corresponding angles.
Triangles
- A triangle is a shape with three vertices, three sides, and three interior angles.
- Types of triangles:
- Scalene triangles: no sides of equal length.
- Isosceles triangles: two sides of equal length.
- Equilateral triangles: all sides of equal length.
- Acute triangles: all angles acute (less than 90 degrees).
- Obtuse triangles: one angle obtuse (greater than 90 degrees).
- Right triangles: one right angle (90 degrees).
Tessellations
- A tessellation is a repeating pattern of shapes that fit together without overlapping.
- Regular tessellations consist of one repeated polygon.
- Semi-regular tessellations consist of two or more types of repeated polygons.
Pythagorean Theorem
- The Pythagorean theorem states that a^2 + b^2 = c^2 in a right triangle.
- The theorem is useful for finding the length of the hypotenuse or a leg in a right triangle.
Angle Relationships
- Angles are formed by two rays drawn from a shared endpoint.
- Angle relationships:
- Complementary angles: add to 90 degrees.
- Supplementary angles: add to 180 degrees.
- Adjacent angles: share a common vertex and side.
- Linear pair: two angles that add to 180 degrees and share a common vertex and side.
- Vertical angles: two angles formed by intersecting lines.
Three-Dimensional Shapes
- A polyhedron is a three-dimensional solid with flat faces.
- A regular polyhedron (Platonic solid) has all faces that are regular polygons that are congruent to each other.
- Convex polyhedra have no vertices or edges that go "into" the polyhedron.
Cones
- A cone is a three-dimensional object with a circular base that narrows to a tip or apex.
- Dimensions:
- Height (h): distance from apex to a point at a right angle to the base.
- Radius (r): distance from center of base to perimeter.
- Slant height (s): distance from perimeter of base to apex.
- Area of base (b): area of circular base in square units.
Probability
- Probability is the likelihood that an event will happen, ranging from 0 to 1.
- Types of probability:
- Simple probability: probability of one event.
- Sequential probability: probability of two or more events.
- Events can be dependent or independent.### Probability in Real-Life Situations
- Probability is widely used in everyday life to predict outcomes, such as weather forecasts and game-winning chances.
Theoretical Probability
- Theoretical probability is a method to express the likelihood of an event occurring.
- It is calculated by dividing the number of favorable outcomes by the total possible outcomes.
- The result is a ratio that can be expressed as a fraction and a probability value between 0 and 1.
- The probability value can be easily converted into a percentage.
When to Use Theoretical Probability
- Theoretical probability is appropriate when:
- The researcher has intimate knowledge of the subject.
- The researcher can determine all possible and all favorable outcomes.
- Direct experimentation is not possible.
Theoretical vs. Experimental Probability
- Theoretical probability relies on logic and thorough knowledge of the subject.
- Experimental probability uses tests and experiments to determine probability.
- Both are branches of mathematics called probability, which addresses the likelihood of an event occurring under certain conditions.
Measuring Probability
- The scale of measuring probability ranges from 0 to 1.
- 0 represents an impossible event.
- 0.25 represents an unlikely event.
- 0.5 represents an even chance of an event occurring.
- 0.75 represents a likely event.
- 1 represents a certain event.
Experimental Probability Formula
- The formula for calculating experimental probability after trials is: [insert formula].
Learn about transformations in math, including reflections, rotations, and translations, as well as non-rigid transformations like shears.
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