Identify the reflection rule on a coordinate plane that verifies that triangle A(-1,7), B(6,5), C(-2,2) and A'(-1,-7), B'(6,-5), C'(-2,-2) triangle are congruent when reflected over the x-axis.
The reflection rule on a coordinate plane states that when a point (x, y) is reflected over the x-axis, its new coordinates become (x, -y).
Using this rule, we can verify that the reflected triangle A'B'C' is congruent to triangle ABC.
For triangle ABC:
Vertex A: (-1, 7) is reflected to vertex A': (-1, -7)
Vertex B: (6, 5) is reflected to vertex B': (6, -5)
Vertex C: (-2, 2) is reflected to vertex C': (-2, -2)
By comparing the corresponding vertices, we can see that the distance and angle measures between the points remain the same after the reflection. Therefore, we can conclude that triangle A'B'C' is congruent to triangle ABC when reflected over the x-axis.
How do I tell what equations are graphed
To determine the equations that are graphed on a coordinate plane, you can follow these steps:
1. Identify the key points on the graph: Look for any distinct points or intersections on the graph that can help you identify the equation(s) being graphed.
2. Determine the shape of the graph: Examine the overall shape of the graph to get an idea of the type of equation it might represent. For example, a straight line could indicate a linear equation, while a curve may suggest a quadratic, exponential, or trigonometric equation.
3. Consider the symmetry: Determine if the graph exhibits any symmetry, such as being symmetric with respect to the x-axis, y-axis, or origin. This can provide additional clues about the equation being graphed.
4. Analyze the slope or rate of change: If the graph is a straight line, find two points on the line and use them to calculate the slope, which can help you deduce the equation of a line.
5. Use intercepts: Identify any x-intercepts (where the graph intersects the x-axis) and y-intercepts (where the graph intersects the y-axis). These intercepts can help you find the equation of the graph.
6. Apply transformations: If the graph appears to be a transformation of a basic function (such as shifting, stretching, or reflecting), consider how these transformations affect the original equation.
By considering these steps, you can make educated guesses about the equations being graphed on a coordinate plane. However, keep in mind that multiple equations could potentially fit the same graph, so further analysis or information may be needed for a definitive identification.
To determine if two triangles are congruent after a reflection over the x-axis, we need to see if the corresponding sides have the same length and the corresponding angles have the same measures. The reflection rule for a point (x, y) is (x, -y).
Now, let's apply the reflection rule to each point of Triangle ABC to find the corresponding points of Triangle A'B'C':
A'(-1, -7) is the reflection of A(-1, 7).
B'(6, -5) is the reflection of B(6, 5).
C'(-2, -2) is the reflection of C(-2, 2).
Now we can compare the corresponding sides and angles:
1. Side AB:
The distance between A(-1, 7) and B(6, 5) is:
AB = √[(6 - (-1))^2 + (5 - 7)^2]
= √[(7)^2 + (-2)^2]
= √[49 + 4]
= √53.
The distance between A'(-1, -7) and B'(6, -5) is:
A'B' = √[(6 - (-1))^2 + (-5 - (-7))^2]
= √[(7)^2 + (2)^2]
= √[49 + 4]
= √53.
Since AB = A'B', the corresponding sides are congruent.
2. Side BC:
Using the same process, you will find that BC = B'C'.
3. Side AC:
Using the same process, you will find that AC = A'C'.
Therefore, all corresponding sides of Triangle ABC and Triangle A'B'C' are congruent, satisfying the congruence condition for triangles. Hence, Triangle ABC and Triangle A'B'C' are congruent after reflection over the x-axis.