Question

draw △lmn with vertices l(2, −2), m(5, −3), and n(4,2). find the coordinates of the vertices after a 90° rotation about the origin and about each of the points l, m, and n. what are the coordinates of the points after a 90° rotation about the origin? l, m, n help me solve this view an example get more help clear all

Explanation:

Step1: Recall 90° rotation rule

For a point \((x, y)\) rotated 90° counterclockwise about the origin, the new coordinates are \((-y, x)\). For clockwise, it's \((y, -x)\). Assuming counterclockwise (standard unless stated), let's use \((x, y) \to (-y, x)\).

Step2: Rotate point L(2, -2)

Apply the rule: \(x = 2\), \(y = -2\). New \(x' = -(-2) = 2\), \(y' = 2\). So \(L' = (2, 2)\)? Wait, no: Wait, 90° counterclockwise: \((x,y) \to (-y, x)\). So \(L(2, -2)\): \(x' = -(-2) = 2\)? Wait, no: \(y\) is -2, so \(-y = 2\), \(x = 2\), so \(L'(2, 2)\)? Wait, no, let's recheck. Wait, 90° counterclockwise: the formula is \((x, y) \mapsto (-y, x)\). So for \(L(2, -2)\): \(x' = -(-2) = 2\), \(y' = 2\). So \(L'(2, 2)\)? Wait, no, maybe I mixed up. Wait, 90° clockwise: \((x, y) \mapsto (y, -x)\). Let's confirm the standard rotation. The standard 90° counterclockwise rotation about the origin transforms \((x, y)\) to \((-y, x)\). So:

• For \(L(2, -2)\): \(x = 2\), \(y = -2\). So \(x' = -y = -(-2) = 2\), \(y' = x = 2\). So \(L'(2, 2)\)? Wait, no, that seems off. Wait, maybe the original points are L(2, -2), M(5, -3), N(4, 2). Let's do each:

Step3: Rotate M(5, -3)

Using 90° counterclockwise: \(x' = -(-3) = 3\), \(y' = 5\). So \(M'(3, 5)\)? Wait, no: \((x, y) \to (-y, x)\). So \(x = 5\), \(y = -3\). So \(x' = -y = -(-3) = 3\), \(y' = x = 5\). So \(M'(3, 5)\).

Step4: Rotate N(4, 2)

Using 90° counterclockwise: \(x' = -2\), \(y' = 4\). So \(N'(-2, 4)\).

Wait, maybe I made a mistake. Let's recall the correct formula: 90° counterclockwise rotation about the origin: \((x, y)
ightarrow (-y, x)\). So:

• \(L(2, -2)\): \(x = 2\), \(y = -2\). So \(-y = 2\), \(x = 2\). So \(L'(2, 2)\)? Wait, no, that would be if \(y\) is -2, then \(-y\) is 2, and \(x\) is 2, so (2, 2). But let's check with a simple point: (1, 0) rotated 90° counterclockwise is (0, 1). Using the formula: \(x=1\), \(y=0\). \(-y=0\), \(x=1\). So (0, 1). Correct. (0, 1) rotated 90° counterclockwise is (-1, 0). Formula: \(x=0\), \(y=1\). \(-y=-1\), \(x=0\). So (-1, 0). Correct. (1, 1) rotated 90° counterclockwise is (-1, 1). Formula: \(x=1\), \(y=1\). \(-y=-1\), \(x=1\). Wait, no: (1,1) rotated 90° counterclockwise is (-1, 1)? Wait, no, (1,1) rotated 90° counterclockwise should be (-1, 1)? Wait, no, let's plot: (1,1) is in Q1. Rotating 90° counterclockwise around origin: the x and y swap, and x becomes negative. So (1,1) → (-1, 1)? Wait, no, (x,y) → (-y, x). So (1,1) → (-1, 1). Yes. So (2, -2): x=2, y=-2. So -y = 2, x=2. So (2, 2). M(5, -3): x=5, y=-3. -y=3, x=5. So (3, 5). N(4, 2): x=4, y=2. -y=-2, x=4. So (-2, 4).

So the coordinates after 90° counterclockwise rotation about the origin are:

• \(L'(2, 2)\)
• \(M'(3, 5)\)
• \(N'(-2, 4)\)

If it's clockwise, the formula is \((x, y)
ightarrow (y, -x)\). Let's check that:

• \(L(2, -2)\): \(y=-2\), \(-x=-2\). So ( -2, -2)? No, wait, clockwise 90°: (x,y) → (y, -x). So (2, -2) → (-2, -2)? Wait, no: (1,0) clockwise 90° is (0, -1). Formula: (1,0) → (0, -1). Correct. (0,1) clockwise 90° is (1, 0). Formula: (0,1) → (1, 0). Correct. (1,1) clockwise 90° is (1, -1). Formula: (1,1) → (1, -1). Correct. So for \(L(2, -2)\) clockwise: ( -2, -2)? Wait, no: \(x=2\), \(y=-2\). So \(y=-2\), \(-x=-2\). So ( -2, -2). M(5, -3): \(y=-3\), \(-x=-5\). So (-3, -5). N(4, 2): \(y=2\), \(-x=-4\). So (2, -4).

But the problem says "90° rotation about the origin" – usually, unless specified, it's counterclockwise. So assuming counterclockwise, the coordinates are \(L'(2, 2)\), \(M'(3, 5)\), \(N'(-2, 4)\).

Answer:

The coordinates after a 90° counterclockwise rotation about the origin are \(L'(2, 2)\), \(M'(3, 5)\), and \(N'(-2, 4)\). (If clockwise, they would be \(L'(-2, -2)\), \(M'(-3, -5)\), \(N'(2, -4)\).)