Question

5 triangle lmn with vertices l(0, 3), m(3, 4), and n(1, 2): a) 180° rotation about the origin b) dilation with scale factor of 2 using the origin as the center

Explanation:

Part (a): 180° Rotation about the Origin

To rotate a point \((x, y)\) 180° about the origin, we use the rule: \((x, y) \to (-x, -y)\).

Step 1: Rotate \( L(0, 3) \)

Applying the rule: \( (0, 3) \to (-0, -3) = (0, -3) \).

Step 2: Rotate \( M(3, 4) \)

Applying the rule: \( (3, 4) \to (-3, -4) \).

Step 3: Rotate \( N(1, 2) \)

Applying the rule: \( (1, 2) \to (-1, -2) \).

Part (b): Dilation with Scale Factor 2 (Origin as Center)

To dilate a point \((x, y)\) with scale factor \( k \) about the origin, we use the rule: \((x, y) \to (kx, ky)\).

Step 1: Dilate \( L(0, 3) \)

Applying the rule: \( (0, 3) \to (2 \cdot 0, 2 \cdot 3) = (0, 6) \).

Step 2: Dilate \( M(3, 4) \)

Applying the rule: \( (3, 4) \to (2 \cdot 3, 2 \cdot 4) = (6, 8) \).

Step 3: Dilate \( N(1, 2) \)

Applying the rule: \( (1, 2) \to (2 \cdot 1, 2 \cdot 2) = (2, 4) \).

Final Answers

a) Rotated vertices: \( L'(0, -3) \), \( M'(-3, -4) \), \( N'(-1, -2) \)
b) Dilated vertices: \( L'(0, 6) \), \( M'(6, 8) \), \( N'(2, 4) \)

Answer:

Part (a): 180° Rotation about the Origin

To rotate a point \((x, y)\) 180° about the origin, we use the rule: \((x, y) \to (-x, -y)\).

Step 1: Rotate \( L(0, 3) \)

Applying the rule: \( (0, 3) \to (-0, -3) = (0, -3) \).

Step 2: Rotate \( M(3, 4) \)

Applying the rule: \( (3, 4) \to (-3, -4) \).

Step 3: Rotate \( N(1, 2) \)

Applying the rule: \( (1, 2) \to (-1, -2) \).

Part (b): Dilation with Scale Factor 2 (Origin as Center)

To dilate a point \((x, y)\) with scale factor \( k \) about the origin, we use the rule: \((x, y) \to (kx, ky)\).

Step 1: Dilate \( L(0, 3) \)

Applying the rule: \( (0, 3) \to (2 \cdot 0, 2 \cdot 3) = (0, 6) \).

Step 2: Dilate \( M(3, 4) \)

Applying the rule: \( (3, 4) \to (2 \cdot 3, 2 \cdot 4) = (6, 8) \).

Step 3: Dilate \( N(1, 2) \)

Applying the rule: \( (1, 2) \to (2 \cdot 1, 2 \cdot 2) = (2, 4) \).

Final Answers

a) Rotated vertices: \( L'(0, -3) \), \( M'(-3, -4) \), \( N'(-1, -2) \)
b) Dilated vertices: \( L'(0, 6) \), \( M'(6, 8) \), \( N'(2, 4) \)