Rigid Transformations and Congruence
Two figures are congruent if one can be mapped exactly onto the other using only these types of transformations.
What are rigid transformations? (translations, reflections, and/or rotations)
A dilation that makes a figure larger has a scale factor that is:
Greater than 1 or Between 0 and 1?
What is Greater than 1?
In a proportional relationship graphed on a coordinate plane, the line always passes through this point.
What is the origin (0, 0)?
The equation 3x + 6 = 3(x + 2) has this many solutions.
What are infinitely many solutions? (Both sides are equivalent.)
A relation is a function if each input has exactly ______ output(s).
What is exactly one output?
This rigid transformation slides every point of a figure the same distance and in the same direction, without flipping or rotating it.
What is a translation?
When a triangle is dilated by a scale factor of 1/2, the side lengths of the image are this fraction of the original.
What is one-half (½) of the original side lengths?
In the equation y = mx + b, this letter represents the y-intercept of the linear relationship.
What is b?
When solving a system of linear equations, the intersection point of the two lines represents the _______ to the system.
What is the solution — the (x, y) pair that satisfies both equations?
A graph passes the vertical line test if no vertical line crosses it more than once. What does passing this test tell you about the graph?
It tells you the graph represents a function — every input (x-value) has exactly one output (y-value).
After performing a rotation, reflection, or translation on a figure, the original figure is called the pre-image, and the result is called the _______.
What is the image?
To show two figures are similar, you must show there is a sequence of these transformations that maps one figure to the other.
What are rigid transformations and dilations? (translations, reflections, rotations, and dilations)
A worker earns $12 more for every hour they work. This value — $12 — represents this feature of the linear relationship.
What is the slope (or rate of change)?
Solve:
5x − 3 = 2x + 9
What is x = 4?
This formula is used to calculate the volume of a cylinder with radius r and height h.
What is V = πr²h?
When two parallel lines are cut by a transversal, rigid transformations can be used to show that these angle pairs are always equal.
What are alternate interior angles? (corresponding angles)
The slope of a line is the same between any two points on that line because all slope triangles drawn on the same line are ________ to each other.
What is similar?
Write an equation for the linear relationship that passes through the points (2, 5) and (4, 9).
What is y = 2x + 1?
A system of two linear equations has no solution. What does this tell you about the graphs of the two equations?
The two lines are parallel — they have the same slope but different y-intercepts, so they never intersect.
The volume of a cone is this fraction, ________, of the volume of a cylinder with the same base and height. The equation for the volume of a cone is ________.
What is one-third (1/3)?
V = ⅓πr²h
A student reflects a triangle across the y-axis and then rotates it 180° about the origin. Explain what stays the same and what could be different about the pre-image and image.
Side lengths and angle measures stay the same (the figures are congruent); the orientation and position of the figure change.
A line passes through (2, 3) and (4, 7). I can use this formula: ________ to calculate the slope which equals ______.
What is m = y2-y1 / x2-x1?
What is 2?
A plumber charges a flat fee of $60 to visit a home, plus an additional $45 per hour for labor. Write a linear equation in slope-intercept form (y=mx+b) representing the total cost (y) for a repair that takes x hours.
Calculate the total cost for a 4-hour job.
y = 45x + 60
$240
Use substitution to solve: y = 2x + 1 and y = −x + 7
What is (2, 5)?
The formula to find volume of a sphere is __________.
A sphere has a radius of 3 cm. Calculate its volume.
What is V = 4/3 πr3
What is 36π cm³?