Question

find the final image points after the sequence of transformations for the triangle below:
pick 2, if you do all 3 youll get extra credit
b(-3, 2) d(2, 3) c(1, 5)
a) rotation 90 degrees cc
b) (x, y) ---> (x +3, y - 7)
c) reflection across the x-axis

Explanation:

Step1: Recall Rotation Rule (90° CCW)

For a point \((x, y)\), rotating 90° counterclockwise transforms it to \((-y, x)\).

For Point B(-3, 2):

\(x = -3\), \(y = 2\). New coordinates: \((-2, -3)\).

For Point D(2, 3):

\(x = 2\), \(y = 3\). New coordinates: \((-3, 2)\).

For Point C(1, 5):

\(x = 1\), \(y = 5\). New coordinates: \((-5, 1)\).

Step2: Recall Translation Rule \((x, y) \to (x + 3, y - 7)\)

For Point B(-3, 2):

\(x + 3 = -3 + 3 = 0\), \(y - 7 = 2 - 7 = -5\). New coordinates: \((0, -5)\).

For Point D(2, 3):

\(x + 3 = 2 + 3 = 5\), \(y - 7 = 3 - 7 = -4\). New coordinates: \((5, -4)\).

For Point C(1, 5):

\(x + 3 = 1 + 3 = 4\), \(y - 7 = 5 - 7 = -2\). New coordinates: \((4, -2)\).

Step3: Recall Reflection Across x - axis Rule

For a point \((x, y)\), reflecting across the x - axis transforms it to \((x, -y)\).

For Point B(-3, 2):

New coordinates: \((-3, -2)\).

For Point D(2, 3):

New coordinates: \((2, -3)\).

For Point C(1, 5):

New coordinates: \((1, -5)\).

(Let's choose part a and part b for demonstration. If we take rotation (a) and translation (b)):

First, apply rotation (a) to get intermediate points, then apply translation (b) to those intermediate points.

Intermediate after rotation (a):

B': \((-2, -3)\), D': \((-3, 2)\), C': \((-5, 1)\)

Apply translation (b) to intermediate points:

For B': \(x=-2 + 3=1\), \(y=-3 - 7=-10\) → \((1, -10)\)

For D': \(x=-3 + 3 = 0\), \(y=2 - 7=-5\) → \((0, -5)\)

For C': \(x=-5 + 3=-2\), \(y=1 - 7=-6\) → \((-2, -6)\)

Answer:

(If we choose transformation a (90° CCW rotation) and b (translation \((x + 3,y - 7)\)):

For point B(-3,2):

• After rotation: \((-2, -3)\)
• After translation on rotated point: \((1, -10)\)

For point D(2,3):

• After rotation: \((-3, 2)\)
• After translation on rotated point: \((0, -5)\)

For point C(1,5):

• After rotation: \((-5, 1)\)
• After translation on rotated point: \((-2, -6)\)

(Or if we choose other combinations, the answer will vary based on the chosen transformations. For example, if we choose a and c:

After rotation (a): B(-2,-3), D(-3,2), C(-5,1)

After reflection (c) on rotated points: B(-2, 3), D(-3, -2), C(-5, -1)

If we choose b and c:

After translation (b): B(0,-5), D(5,-4), C(4,-2)

After reflection (c) on translated points: B(0,5), D(5,4), C(4,2))

(The answer depends on the two chosen transformations. Let's assume we choose a and b as above, the final image points for B, D, C are \((1, -10)\), \((0, -5)\), \((-2, -6)\) respectively)