Congruence
Two figures have the same size and the same shape. What word describes these figures?
What is congruent?
This transformation slides a figure left, right, up, or down without changing its size or shape.
What is a translation?
What makes an equation linear?
Answer: The variable has a degree of 1, constant rate of change, and its graph is a straight line.
Two linear equations are graphed on the same coordinate plane. The lines cross at exactly one point.
What does the point where they intersect represent?
The solution to the system — the ordered pair that makes both equations true
Triangle ABC is congruent to Triangle DEF. If ∠A = 50°, what is the measure of ∠D?
What is 50°?
A figure is flipped over a line so that every point is the same distance from the line as its corresponding point on the other side.
What type of transformation is this?
Reflection
In the equation y = 4x - 7, what does the 4 tell you about the line?
It is the slope, which tells how steep the line is and how it changes from left to right.
A system has two lines that are parallel and never intersect.
How many solutions does the system have?
No solution
Triangle ABC is congruent to Triangle XYZ. If AB = 8 cm, which side of Triangle XYZ must also measure 8 cm?
What is XY?
A figure is turned around a fixed point. The figure stays the same size and shape, but its orientation changes.
What type of transformation is this?
Two lines have the same slope but different y-intercepts. What do you know about these two lines?
They are parallel and will never intersect.
Two equations are graphed and both equations create the exact same line.
How many solutions does the system have?
Infinitely many solutions
A triangle is translated 5 units right and 3 units up. Is the new triangle congruent to the original? Explain why.
What is yes, because a translation preserves size and shape?
A student says, “A reflection changes the size of a figure because the figure is flipped.”
Is the student correct? Explain your reasoning.
No. A reflection changes the position/orientation of a figure but preserves its size and shape.
A student says, “The equation y = -2x + 6 has a positive slope because the 6 is positive.”
Is the student correct? Explain.
No. The slope is −2. The 6 is the y-intercept, not the slope.
A student solves a system and gets the answer (3,5)(3, 5).
What does this mean about the two original equations?
Answer: When x=3 and y=5, both equations are true. On a graph, the two lines intersect at (3,5)(3,5).
Triangle ABC has vertices A(1,2), B(4,2), and C(1,6). It is translated 3 units right and 2 units down. What are the coordinates of A′, B′, and C′?
What are A′(4,0), B′(7,0), and C′(4,4)?
A triangle is first translated and then reflected. After both transformations, the triangle is still the same size and shape as the original.
What property of transformations explains why the original triangle and final triangle are congruent?
Translations and reflections preserve distance and angle measures, so they preserve congruence.
Two different equations represent the same line.
What must be true about their graphs, and what does that tell you about their solutions?
Their graphs are the same line, so they have infinitely many solutions in common.
Without solving the system, determine how many solutions it has:
y=2x+4
2y=4x+8
Infinitely many solutions, because the second equation simplifies to y=2x+4, which is the exact same line as the first equation.