Trapezoid JKLM has points J(-2,4), K(2,-4), L(4,2), and M(-4,2). Which transformation will map the trapezoid to itself? a. reflection across the x-axis b. reflection across the y-axis*** c. rotation of 90 degree about the origin and d. rotation of 180 degrees about the origin
If you write out the points, then provide the mapping rules, you can check each new rule...
You may have missed a negative sign in your original stating of the points :)
To determine which transformation will map the trapezoid to itself, we need to examine the properties of each transformation and their effect on the coordinates of the given points.
a. Reflection across the x-axis:
A reflection across the x-axis will flip the trapezoid vertically. It will swap the y-coordinates of each point while keeping the x-coordinates the same.
Applying this transformation to the given trapezoid:
J(-2,4) -> J'(-2,-4)
K(2,-4) -> K'(2,4)
L(4,2) -> L'(4,-2)
M(-4,2) -> M'(-4,-2)
As we can see, the trapezoid is not mapped to itself after this transformation, so option a is not correct.
b. Reflection across the y-axis:
A reflection across the y-axis will flip the trapezoid horizontally. It will swap the x-coordinates of each point while keeping the y-coordinates the same.
Applying this transformation to the given trapezoid:
J(-2,4) -> J'(2,4)
K(2,-4) -> K'(-2,-4)
L(4,2) -> L'(-4,2)
M(-4,2) -> M'(4,2)
Here, we can see that the trapezoid is indeed mapped to itself after this transformation. Therefore, option b is correct.
c. Rotation of 90 degrees about the origin:
A rotation of 90 degrees about the origin will move each point to a new position by rotating it counterclockwise by 90 degrees. The new coordinates (x', y') can be determined using the formulas:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
Applying this transformation to the given trapezoid and using θ = 90 degrees:
J(-2,4) -> J'(-4,-2)
K(2,-4) -> K'(4,-2)
L(4,2) -> L'(-2,-4)
M(-4,2) -> M'(2,4)
As we can see, the trapezoid is not mapped to itself after this transformation, so option c is not correct.
d. Rotation of 180 degrees about the origin:
A rotation of 180 degrees about the origin will move each point to a new position by rotating it counterclockwise by 180 degrees. The new coordinates (x', y') can be determined using the formulas:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
Applying this transformation to the given trapezoid and using θ = 180 degrees:
J(-2,4) -> J'(2,-4)
K(2,-4) -> K'(-2,4)
L(4,2) -> L'(-4,-2)
M(-4,2) -> M'(4,-2)
Here, we can see that the trapezoid is not mapped to itself after this transformation, so option d is not correct.
In conclusion, the correct transformation that will map the trapezoid to itself is a reflection across the y-axis, which is option b.