Answer
The coordinates of the vertices of KLM after all the given transformation steps are K''''(-18, 9), L''''(-30, -15), M''''(12, -21)
Explanation
We need to perform a series of transformation on a triangle KLM whose coordinates are given as K (4,2), L(-4,6), M(-6,-8)
The transformations include
- 90 degree clockwise rotation.
- A transformation of right 4 units and up 1 unit.
- A reflection in the origin,
- and a dilation with a scale factor of 3.
We will take each of them one at a time
- 90 degree clockwise rotation.
In transforming a point (x, y) by rotating it 90 degrees clockwise, the new coordinates are given as (y, -x). That is, we change the coordinates and then add minus to the x, which is now the y-coordinate. So, for our triangle,
K (4,2), L(-4,6), M(-6,-8)
K' = (2, -4)
L' = (6, 4)
M' = (-8, 6)
The coordinates after the first step is
K'(2, -4), L'(6, 4), M'(-8, 6)
- A transformation of right 4 units and up 1 unit.
This is a simple transformation, a transformation to the right adds that number of units to the x-coordinate and a transformation up adds that number of units to the y-coordinate. So, if the original coordinate is (x, y), translating to the right by 4 units and up by 1 unit turns the new coordinates to (x+4, y+1). So, after the first step, we had K'(2, -4), L'(6, 4), M'(-8, 6), the new coordinates will now be
K'' = (2+4, -4+1) = (6, -3)
L'' = (6+4, 4+1) = (10, 5)
M'' = (-8+4, 6+1) = (-4, 7)
The coordinates after the second step is
K''(6, -3), L''(10, 5), M''(-4, 7)
- A reflection in the origin
When a point is reflected in the origin, both the x and y coordinates have their signs changed. That is, coordinates (x, y) become (-x, -y). So, our points from the second step, K''(6, -3), L''(10, 5), M''(-4, 7) will have new coordinates
K''' = (-6, 3)
L''' = (-10, -5)
M''' = (4, -7)
The coordinates after the third step
K'''(-6, 3), L'''(-10, -5), M'''(4, -7)
- And a dilation with a scale factor of 3.
A dilation means the size is increased or decreased. If the scale factor is less than 1, then the size is decreased, but if the scale factor is more than 1, it means the figure is enlarged.
For our question, we want to dilate with a scale factor of 3, meaning we want to enlarge the size of our triangle by 3.
To dilate a figure by a scale factor, we need to first pick a reference point, the question already told us to dilate about the origin, which makes it all very easy for us.
Dilating about the origin just multiplies the coordinates by the scale factor. So, dilating (x, y) about the origin by a scale factor k, gives new coordinates (kx, ky). So, our coordinates K'''(-6, 3), L'''(-10, -5), M'''(4, -7) when dilated by a scale factor of 3 about the origin become
K'''' = (-18, 9)
L'''' = (-30, -15)
M'''' = (12, -21)
The coordinates of the triangle after the final step are
K''''(-18, 9), L''''(-30, -15), M''''(12, -21)
Hope this Helps!!!