What are the coordinates of the point (-9, 4) after a 90 degree rotation followed by a translation 3 units right?

The coordinates of the point (-9, 4) after a 90 degree rotation followed by a translation 3 units right are (8,9) (for clockwise rotation) and (0,-9) (for counterclockwise rotation)

How does rotation by 90 degrees changes coordinates of a point if rotation is with respect to origin?

Let the point be having coordinates (x,y).

Case 1: If the point is in first quadrant:

Subcase: Clockwise rotation:

Then (x,y) → (y, -x)

Subcase: Counterclockwise rotation:

Then (x,y) → (-y, x)

Case 2: If the point is in second quadrant:

Subcase: Clockwise rotation:

Then (x,y) → (y, -x)

Subcase: Counterclockwise rotation:

Then (x,y) → (-y, x)

Case 3: If the point is in third quadrant:

Subcase: Clockwise rotation:

Then (x,y) → (y, -x)

Subcase: Counterclockwise rotation:

Then (x,y) → (-y, x)

Case 4: If the point is in fourth quadrant:

Subcase: Clockwise rotation:

Then (x,y) → (y, -x)

Subcase: Counterclockwise rotation:

Then (x,y) → (-y, x)

Case 5: For points on axes

You can take that point in any of the two surrounding quadrants. Example, if the point is on positive x axis, then it can taken as of first quadrant or fourth quadrant.

Case 6: On origin

No effect as we assumed rotation is being with respect to origin.

For this case, the point (-9,4) (present in second quadrant) is rotated by 90 degrees.

This rotation can be either clockwise, or counterclockwise.

Thus, for each case, we have:

case 1: Clockwise rotation:

Then (x,y) → (y, -x), therefore, (-9,4) → (4, 9)

case 2: Counterclockwise rotation:

Then (x,y) → (-y, x), therefore, (-9,4) → (-4, -9)

Now, when shifting to right by 4, that means, we're making no changes vertically, but old x-coordinates are increased by 4 to make them shift to right side as value of x increases as we go more and more to the right side with respect to the origin and in defaultly oriented cartesian space.

Thus, we have:

case 1: Clockwise rotation:

After translation of 4 units right:

(4, 9) → (4+4, 9) = (8,9)

case 2: Counterclockwise rotation:

After translation of 4 units right:

(-4, -9) → (-4+4, -9) = (0,-9)

Thus, the coordinates of the point (-9, 4) after a 90 degree rotation followed by a translation 3 units right are (8,9) (for clockwise rotation) and (0,-9) (for counterclockwise rotation)

Learn more about rotation of a point with respect to origin here:

https://brainly.com/question/18856342