Triangle ABC has vertices A(–4,–2), B(–1,3), and C(5,0). Triangle ABC is rotated 180° counterclockwise about the origin to form triangle A′B′C′.
What are the coordinates of the vertices of the triangle A′B′C′?
How could you determine the coordinates of the vertices of the image without using a graph?
3.c
4.c
5.d
6.a
7.c
8.d
9.b
11. b, d and e
13.d
The rest I cannot help with
A rotation of 180° is a reflection in the origin , so just change the signs of all your points.
e.g. B(-1,3) ----> B'(1,-3)
etc
To find the coordinates of the vertices of triangle A'B'C', we need to rotate each individual point counterclockwise by 180 degrees about the origin.
Let's start with point A(-4, -2):
To rotate the point (x, y) counterclockwise by 180 degrees, we need to multiply the x-coordinate by -1 and the y-coordinate by -1.
For point A: A'(-x, -y) = (-(-4), -(-2)) = (4, 2)
Next, let's rotate point B(-1, 3):
For point B: B'(-x, -y) = (-(-1), -3) = (1, -3)
Finally, let's rotate point C(5, 0):
For point C: C'(-x, -y) = (-(5), -(0)) = (-5, 0)
Therefore, the coordinates of the vertices of triangle A'B'C' are:
A' (4, 2)
B' (1, -3)
C' (-5, 0)
To determine the coordinates of the vertices of the image without using a graph, we used the concept of rotating points counterclockwise by 180 degrees about the origin. By applying the rotation rule, we calculated the new coordinates of each point A', B', and C'.
To find the coordinates of the new vertices A′, B′, and C′, we need to rotate each of the original coordinates (A, B, and C) counterclockwise by 180 degrees about the origin.
To do this, we can use the following rotation formula:
For a point P(x, y) rotated counterclockwise by θ degrees about the origin, the new coordinates (x', y') can be found using the following formulas:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
In this case, since we are rotating by 180 degrees (θ = 180), the formulas simplify to:
x' = -x
y' = -y
Now, let's apply these formulas to find the coordinates of A′, B′, and C′.
For A(–4,–2):
x' = -(-4) = 4
y' = -(-2) = 2
So, A′ is located at (4, 2).
For B(–1,3):
x' = -(-1) = 1
y' = -(3) = -3
So, B′ is located at (1, -3).
For C(5,0):
x' = -(5) = -5
y' = -(0) = 0
So, C′ is located at (-5, 0).
Therefore, the coordinates of the vertices of triangle A′B′C′ are A′ (4, 2), B′ (1, -3), and C′ (-5, 0).
To determine the coordinates of the vertices without using a graph, we simply applied the rotation formula for counterclockwise rotation by 180 degrees. By plugging in the original coordinates into the formulas, we can calculate the new coordinates directly. This method allows us to find the image without relying on a visual graph.