Question

6. Triangle ABC is rotated 90 degrees counterclockwise and then translated 3 units up to form

Triangle

ABC: Which transformation can be used to map each point (.y) on Triangle ABC

to its corresponding point on Triangle A B C?

A

G.y) - (y + 3, x)

B

6. y) - (x+3)

(x, y) - (y+3.x)

D. (.y)-( y. x+3)

C

Answer 1

Given:

Consider triangle ABC is rotated 90 degrees counterclockwise and then translated 3 units up to form triangle A'B'C'.

To find:

The transformation that can be used to map each point (x,y) on Triangle ABC to its corresponding point on Triangle A'B'C'.

Solution:

If a figure rotated 90 degrees counterclockwise, then

$(x,y)\to (-y,x)$

$P(x,y)\to P_1(-y,x)$

If a figure translated 3 units up, then

$(x,y)\to (x,y+3)$

$P_1(-y,x)\to P'(-y,x+3)$

If  triangle ABC is rotated 90 degrees counterclockwise and then translated 3 units up to form triangle A'B'C', then the rule of transformation is

$(x,y)\to (-y,x+3)$

Therefore, the required rule of transformation is $(x,y)\to (-y,x+3)$.

Note: Option are not in proper form.

Answer 2

All the points of the triangle ABC(assuming it is in first quadrant) of the form (x,y) turns (-x,y+3) after applying considered transformations.

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) → (x, -y)

Subcase: Counterclockwise rotation:

Then (x,y) → (-x, y)

• Case 2: If the point is in second quadrant:

Subcase: Clockwise rotation:

Then (x,y) → (-x, y)

Subcase: Counterclockwise rotation:

Then (x,y) → (x, -y)

• Case 3: If the point is in third quadrant:

Subcase: Clockwise rotation:

Then (x,y) → (x, -y)

Subcase: Counterclockwise rotation:

Then (x,y) → (-x, y)

• Case 4: If the point is in fourth quadrant:

Subcase: Clockwise rotation:

Then (x,y) → (-x, y)

Subcase: Counterclockwise rotation:

Then (x,y) → (x, -y)

• 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.

Since no quadrant is specified, we assume triangle is fully in first quadrant. That means all points (x,y) will turn (-x,y) after counter clockwise rotation.

Now translation of 3 units up means all y coordinates will be increased by 3, thus, all (x,y) points of triangle first become (-x,y) then (-x, y+3).

Thus, all the points of the triangle ABC(assuming it is in first quadrant) of the form (x,y) turns (-x,y+3) after applying considered transformations.

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