Question

rotations (2023)
graph the image of the figure using the transformation given.

1. rotation 180° about the origin
2. rotation 90° counterclockwise about the origin
3. rotation 90° counterclockwise about the origin
4. rotation 180° about the origin
5. rotation 90° counterclockwise about the origin
6. rotation 180° about the origin

Explanation:

To solve these rotation problems, we use the following rules for rotating a point \((x, y)\) about the origin:

Rotation Rules:

• \(180^\circ\) rotation: \((x, y) \to (-x, -y)\)
• \(90^\circ\) counterclockwise rotation: \((x, y) \to (-y, x)\)

Problem 1: Rotation \(180^\circ\) about the origin

1. Identify coordinates of vertices (e.g., \(T(-5, 0)\), \(U(2, 0)\), \(V(2, -3)\), \(S(-4, -2)\)).
2. Apply \(180^\circ\) rule:

• \(T(-5, 0) \to T'(5, 0)\)
• \(U(2, 0) \to U'(-2, 0)\)
• \(V(2, -3) \to V'(-2, 3)\)
• \(S(-4, -2) \to S'(4, 2)\)

1. Plot \(T', U', V', S'\) and connect to form the image.

Problem 2: Rotation \(90^\circ\) counterclockwise about the origin

1. Identify coordinates (e.g., \(Z(-3, 0)\), \(Y(-3, 1)\), \(X(2, 0)\), \(W(2, -3)\)).
2. Apply \(90^\circ\) counterclockwise rule:

• \(Z(-3, 0) \to Z'(0, -3)\)
• \(Y(-3, 1) \to Y'(-1, -3)\)
• \(X(2, 0) \to X'(0, 2)\)
• \(W(2, -3) \to W'(3, 2)\)

1. Plot \(Z', Y', X', W'\) and connect.

Problem 3: Rotation \(90^\circ\) counterclockwise about the origin

1. Identify coordinates (e.g., \(Q(-2, -3)\), \(R(0, -1)\), \(S(2, -1)\), \(T(3, -3)\)).
2. Apply \(90^\circ\) counterclockwise rule:

• \(Q(-2, -3) \to Q'(3, -2)\)
• \(R(0, -1) \to R'(1, 0)\)
• \(S(2, -1) \to S'(1, 2)\)
• \(T(3, -3) \to T'(3, 3)\)

1. Plot \(Q', R', S', T'\) and connect.

Problem 4: Rotation \(180^\circ\) about the origin

1. Identify coordinates (e.g., \(N(3, -2)\), \(L(4, -2)\), \(M(4, -1)\)).
2. Apply \(180^\circ\) rule:

• \(N(3, -2) \to N'(-3, 2)\)
• \(L(4, -2) \to L'(-4, 2)\)
• \(M(4, -1) \to M'(-4, 1)\)

1. Plot \(N', L', M'\) and connect.

Problem 5: Rotation \(90^\circ\) counterclockwise about the origin

1. Identify coordinates (e.g., \(I(-4, 0)\), \(L(-1, 0)\), \(K(-1, 3)\), \(J(-3, 2)\)).
2. Apply \(90^\circ\) counterclockwise rule:

• \(I(-4, 0) \to I'(0, -4)\)
• \(L(-1, 0) \to L'(0, -1)\)
• \(K(-1, 3) \to K'(-3, -1)\)
• \(J(-3, 2) \to J'(-2, -3)\)

1. Plot \(I', L', K', J'\) and connect.

Problem 6: Rotation \(180^\circ\) about the origin

1. Identify coordinates (e.g., \(I(-1, 0)\), \(J(0, 1)\), \(K(2, -1)\), \(H(1, -3)\)).
2. Apply \(180^\circ\) rule:

• \(I(-1, 0) \to I'(1, 0)\)
• \(J(0, 1) \to J'(0, -1)\)
• \(K(2, -1) \to K'(-2, 1)\)
• \(H(1, -3) \to H'(-1, 3)\)

1. Plot \(I', J', K', H'\) and connect.

Final Answer

For each problem, plot the transformed vertices using the rotation rules and connect them to form the image. The key is to apply the correct coordinate transformation for \(180^\circ\) or \(90^\circ\) counterclockwise rotation.

Answer:

To solve these rotation problems, we use the following rules for rotating a point \((x, y)\) about the origin:

Rotation Rules:

• \(180^\circ\) rotation: \((x, y) \to (-x, -y)\)
• \(90^\circ\) counterclockwise rotation: \((x, y) \to (-y, x)\)

Problem 1: Rotation \(180^\circ\) about the origin

1. Identify coordinates of vertices (e.g., \(T(-5, 0)\), \(U(2, 0)\), \(V(2, -3)\), \(S(-4, -2)\)).
2. Apply \(180^\circ\) rule:

• \(T(-5, 0) \to T'(5, 0)\)
• \(U(2, 0) \to U'(-2, 0)\)
• \(V(2, -3) \to V'(-2, 3)\)
• \(S(-4, -2) \to S'(4, 2)\)

1. Plot \(T', U', V', S'\) and connect to form the image.

Problem 2: Rotation \(90^\circ\) counterclockwise about the origin

1. Identify coordinates (e.g., \(Z(-3, 0)\), \(Y(-3, 1)\), \(X(2, 0)\), \(W(2, -3)\)).
2. Apply \(90^\circ\) counterclockwise rule:

• \(Z(-3, 0) \to Z'(0, -3)\)
• \(Y(-3, 1) \to Y'(-1, -3)\)
• \(X(2, 0) \to X'(0, 2)\)
• \(W(2, -3) \to W'(3, 2)\)

1. Plot \(Z', Y', X', W'\) and connect.

Problem 3: Rotation \(90^\circ\) counterclockwise about the origin

1. Identify coordinates (e.g., \(Q(-2, -3)\), \(R(0, -1)\), \(S(2, -1)\), \(T(3, -3)\)).
2. Apply \(90^\circ\) counterclockwise rule:

• \(Q(-2, -3) \to Q'(3, -2)\)
• \(R(0, -1) \to R'(1, 0)\)
• \(S(2, -1) \to S'(1, 2)\)
• \(T(3, -3) \to T'(3, 3)\)

1. Plot \(Q', R', S', T'\) and connect.

Problem 4: Rotation \(180^\circ\) about the origin

1. Identify coordinates (e.g., \(N(3, -2)\), \(L(4, -2)\), \(M(4, -1)\)).
2. Apply \(180^\circ\) rule:

• \(N(3, -2) \to N'(-3, 2)\)
• \(L(4, -2) \to L'(-4, 2)\)
• \(M(4, -1) \to M'(-4, 1)\)

1. Plot \(N', L', M'\) and connect.

Problem 5: Rotation \(90^\circ\) counterclockwise about the origin

1. Identify coordinates (e.g., \(I(-4, 0)\), \(L(-1, 0)\), \(K(-1, 3)\), \(J(-3, 2)\)).
2. Apply \(90^\circ\) counterclockwise rule:

• \(I(-4, 0) \to I'(0, -4)\)
• \(L(-1, 0) \to L'(0, -1)\)
• \(K(-1, 3) \to K'(-3, -1)\)
• \(J(-3, 2) \to J'(-2, -3)\)

1. Plot \(I', L', K', J'\) and connect.

Problem 6: Rotation \(180^\circ\) about the origin

1. Identify coordinates (e.g., \(I(-1, 0)\), \(J(0, 1)\), \(K(2, -1)\), \(H(1, -3)\)).
2. Apply \(180^\circ\) rule:

• \(I(-1, 0) \to I'(1, 0)\)
• \(J(0, 1) \to J'(0, -1)\)
• \(K(2, -1) \to K'(-2, 1)\)
• \(H(1, -3) \to H'(-1, 3)\)

1. Plot \(I', J', K', H'\) and connect.

Final Answer

For each problem, plot the transformed vertices using the rotation rules and connect them to form the image. The key is to apply the correct coordinate transformation for \(180^\circ\) or \(90^\circ\) counterclockwise rotation.