Question
A rotation maps a triangle with vertices A (3,1) , B (-1,-1) , and C (7,-2) to 4A'B'C". What is the length of B'C*, to the nearest unit?
(1 point)
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(1 point)
04
05
08
09
Answers
GPT 3.5
To find the length of B'C*, we need to calculate the distance between points B' and C'.
First, let's find the coordinates of B' and C'.
Since point A is rotated to point A', we know that points B and C will also be rotated by the same angle. Let's assume the triangle is rotated counterclockwise.
To find the coordinates of B', we can use the formula for rotating a point (x, y) counterclockwise by an angle θ:
x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)
Let's assume that point A is rotated to point A' at coordinates (x', y').
Since point A has coordinates (3, 1), we can plug those values in:
x' = 3*cos(θ) - 1*sin(θ)
y' = 3*sin(θ) + 1*cos(θ)
Similarly, we can find the coordinates of B' and C':
x'' = -1*cos(θ) - (-1)*sin(θ)
y'' = -1*sin(θ) + (-1)*cos(θ)
x''' = 7*cos(θ) - (-2)*sin(θ)
y''' = 7*sin(θ) + (-2)*cos(θ)
Now, let's find the values of θ. Since point A is rotated to point A', we know that the initial angle is the angle between the positive x-axis and the line connecting points A and A'. Using the coordinates of A(3, 1) and A', we can find this angle using the arctan function:
θ = atan((y' - 1)/(x' - 3))
Similarly, we can find the angles for points B and C:
θ' = atan((y'' + 1)/(-1 - (-1)))
θ'' = atan((y''' + 2)/(7 - (-2)))
Now, let's calculate the values of θ, θ', and θ'' using the given coordinates:
θ = atan((y' - 1)/(x' - 3))
θ = atan((? - 1)/(? - 3))
θ' = atan((y'' + 1)/(-1 - (-1)))
θ' = atan((? + 1)/(-2))
θ'' = atan((y''' + 2)/(7 - (-2)))
θ'' = atan((? + 2)/(? + 2))
We can use these values of θ, θ', and θ'' to find the coordinates of B' and C'. However, since we don't have the values of θ, θ', and θ'', we cannot calculate the exact coordinates of B' and C'.
Therefore, we cannot find the exact length of B'C*.
First, let's find the coordinates of B' and C'.
Since point A is rotated to point A', we know that points B and C will also be rotated by the same angle. Let's assume the triangle is rotated counterclockwise.
To find the coordinates of B', we can use the formula for rotating a point (x, y) counterclockwise by an angle θ:
x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)
Let's assume that point A is rotated to point A' at coordinates (x', y').
Since point A has coordinates (3, 1), we can plug those values in:
x' = 3*cos(θ) - 1*sin(θ)
y' = 3*sin(θ) + 1*cos(θ)
Similarly, we can find the coordinates of B' and C':
x'' = -1*cos(θ) - (-1)*sin(θ)
y'' = -1*sin(θ) + (-1)*cos(θ)
x''' = 7*cos(θ) - (-2)*sin(θ)
y''' = 7*sin(θ) + (-2)*cos(θ)
Now, let's find the values of θ. Since point A is rotated to point A', we know that the initial angle is the angle between the positive x-axis and the line connecting points A and A'. Using the coordinates of A(3, 1) and A', we can find this angle using the arctan function:
θ = atan((y' - 1)/(x' - 3))
Similarly, we can find the angles for points B and C:
θ' = atan((y'' + 1)/(-1 - (-1)))
θ'' = atan((y''' + 2)/(7 - (-2)))
Now, let's calculate the values of θ, θ', and θ'' using the given coordinates:
θ = atan((y' - 1)/(x' - 3))
θ = atan((? - 1)/(? - 3))
θ' = atan((y'' + 1)/(-1 - (-1)))
θ' = atan((? + 1)/(-2))
θ'' = atan((y''' + 2)/(7 - (-2)))
θ'' = atan((? + 2)/(? + 2))
We can use these values of θ, θ', and θ'' to find the coordinates of B' and C'. However, since we don't have the values of θ, θ', and θ'', we cannot calculate the exact coordinates of B' and C'.
Therefore, we cannot find the exact length of B'C*.