Triangle ABC has vertices A (-4,-2), B (-1,3), and C (5,0). Triangle ABC is rotated 180 degrees counterclockwise about the origin to form triangle A'B'C'.
a. what are the coordinates of the vertices of triangle A'B'C'?
b. How could you determine the coordinates of the vertices of the image without using a graph?
a. A (4,2), B (1,-3), and C (-5,0)
b. as you can see , the sign of every x and y is changed
a. To find the coordinates of triangle A'B'C', we need to rotate each vertex of triangle ABC by 180 degrees counterclockwise about the origin.
1. For vertex A (-4, -2):
The formula to rotate a point (x, y) counterclockwise by 180 degrees about the origin is:
(x', y') = (-x, -y)
Substituting the values for A, we get:
(-4, -2) -> (-(-4), -(-2)) = (4, 2)
Therefore, the coordinates of A' are (4, 2).
2. For vertex B (-1, 3):
Applying the same rotation formula:
(-1, 3) -> (-(-1), -3) = (1, -3)
Therefore, the coordinates of B' are (1, -3).
3. For vertex C (5, 0):
Again, using the rotation formula:
(5, 0) -> (-5, 0)
Therefore, the coordinates of C' are (-5, 0).
So, the coordinates of the vertices of triangle A'B'C' are A' (4, 2), B' (1, -3), and C' (-5, 0).
b. To determine the coordinates without using a graph, you can use the rotation formula mentioned above. By applying the formula to each vertex of the original triangle ABC, you can calculate the coordinates of the corresponding vertices of triangle A'B'C'.
a. To find the coordinates of triangle A'B'C', we first need to understand the concept of rotating a point about the origin. When a point (x, y) is rotated counterclockwise by 180 degrees about the origin, the new coordinates (x', y') can be found using the following formulas:
x' = -x
y' = -y
Using these formulas, we can calculate the coordinates of each vertex:
For vertex A (-4, -2):
x' = -(-4) = 4
y' = -(-2) = 2
So the coordinates of A' are (4, 2).
For vertex B (-1, 3):
x' = -(-1) = 1
y' = -3 = -3
So the coordinates of B' are (1, -3).
For vertex C (5, 0):
x' = -5 = -5
y' = -0 = 0
So the coordinates of C' are (-5, 0).
Therefore, the coordinates of triangle A'B'C' are A'(4, 2), B'(1, -3), and C'(-5, 0).
b. To determine the coordinates of the vertices without using a graph, we can directly apply the rotation formulas mentioned above. We simply substitute the original coordinates of each vertex into these formulas to get the new coordinates. Here's the steps you can follow:
1. Start with the original coordinates of each vertex.
Vertex A: (-4, -2)
Vertex B: (-1, 3)
Vertex C: (5, 0)
2. Apply the rotation formulas to find the new coordinates.
Vertex A':
x' = -(-4) = 4
y' = -(-2) = 2
A' = (4, 2)
Vertex B':
x' = -(-1) = 1
y' = -3
B' = (1, -3)
Vertex C':
x' = -5
y' = 0
C' = (-5, 0)
3. The new coordinates represent the vertices of triangle A'B'C'.
So, by applying the rotational formulas to the original coordinates of each vertex, you can determine the coordinates of the image triangle A'B'C' without using a graph.