Question

1. rhombus pqrs with vertices p(-1, -3), q(3, -4), r(4, -8), and s(0, -7):

a) translation along the vector ⟨4, 10⟩
b) 90° counterclockwise rotation about the origin
p(__) q() r() s(__)

Explanation:

Part a) Translation along vector \(\langle 4, 10

angle\)
To translate a point \((x, y)\) along the vector \(\langle a, b
angle\), we use the rule \((x + a, y + b)\).

Step 1: Translate point \(P(-1, -3)\)

Using the translation rule: \(x = -1 + 4 = 3\), \(y = -3 + 10 = 7\). So \(P' = (3, 7)\).

Step 2: Translate point \(Q(3, -4)\)

Using the translation rule: \(x = 3 + 4 = 7\), \(y = -4 + 10 = 6\). So \(Q' = (7, 6)\).

Step 3: Translate point \(R(4, -8)\)

Using the translation rule: \(x = 4 + 4 = 8\), \(y = -8 + 10 = 2\). So \(R' = (8, 2)\).

Step 4: Translate point \(S(0, -7)\)

Using the translation rule: \(x = 0 + 4 = 4\), \(y = -7 + 10 = 3\). So \(S' = (4, 3)\).

Part b) \(90^\circ\) counterclockwise rotation about the origin

The rule for a \(90^\circ\) counterclockwise rotation about the origin is \((x, y) \to (-y, x)\).

Step 1: Rotate point \(P(-1, -3)\)

Using the rotation rule: \(x = -(-3) = 3\), \(y = -1\). So \(P' = (3, -1)\).

Step 2: Rotate point \(Q(3, -4)\)

Using the rotation rule: \(x = -(-4) = 4\), \(y = 3\). So \(Q' = (4, 3)\).

Step 3: Rotate point \(R(4, -8)\)

Using the rotation rule: \(x = -(-8) = 8\), \(y = 4\). So \(R' = (8, 4)\).

Step 4: Rotate point \(S(0, -7)\)

Using the rotation rule: \(x = -(-7) = 7\), \(y = 0\). So \(S' = (7, 0)\).

Final Answers

Part a)

• \(P' = \boldsymbol{(3, 7)}\)
• \(Q' = \boldsymbol{(7, 6)}\)
• \(R' = \boldsymbol{(8, 2)}\)
• \(S' = \boldsymbol{(4, 3)}\)

Part b)

• \(P' = \boldsymbol{(3, -1)}\)
• \(Q' = \boldsymbol{(4, 3)}\)
• \(R' = \boldsymbol{(8, 4)}\)
• \(S' = \boldsymbol{(7, 0)}\)

Answer:

Part a) Translation along vector \(\langle 4, 10

angle\)
To translate a point \((x, y)\) along the vector \(\langle a, b
angle\), we use the rule \((x + a, y + b)\).

Step 1: Translate point \(P(-1, -3)\)

Using the translation rule: \(x = -1 + 4 = 3\), \(y = -3 + 10 = 7\). So \(P' = (3, 7)\).

Step 2: Translate point \(Q(3, -4)\)

Using the translation rule: \(x = 3 + 4 = 7\), \(y = -4 + 10 = 6\). So \(Q' = (7, 6)\).

Step 3: Translate point \(R(4, -8)\)

Using the translation rule: \(x = 4 + 4 = 8\), \(y = -8 + 10 = 2\). So \(R' = (8, 2)\).

Step 4: Translate point \(S(0, -7)\)

Using the translation rule: \(x = 0 + 4 = 4\), \(y = -7 + 10 = 3\). So \(S' = (4, 3)\).

Part b) \(90^\circ\) counterclockwise rotation about the origin

The rule for a \(90^\circ\) counterclockwise rotation about the origin is \((x, y) \to (-y, x)\).

Step 1: Rotate point \(P(-1, -3)\)

Using the rotation rule: \(x = -(-3) = 3\), \(y = -1\). So \(P' = (3, -1)\).

Step 2: Rotate point \(Q(3, -4)\)

Using the rotation rule: \(x = -(-4) = 4\), \(y = 3\). So \(Q' = (4, 3)\).

Step 3: Rotate point \(R(4, -8)\)

Using the rotation rule: \(x = -(-8) = 8\), \(y = 4\). So \(R' = (8, 4)\).

Step 4: Rotate point \(S(0, -7)\)

Using the rotation rule: \(x = -(-7) = 7\), \(y = 0\). So \(S' = (7, 0)\).

Final Answers

Part a)

• \(P' = \boldsymbol{(3, 7)}\)
• \(Q' = \boldsymbol{(7, 6)}\)
• \(R' = \boldsymbol{(8, 2)}\)
• \(S' = \boldsymbol{(4, 3)}\)

Part b)

• \(P' = \boldsymbol{(3, -1)}\)
• \(Q' = \boldsymbol{(4, 3)}\)
• \(R' = \boldsymbol{(8, 4)}\)
• \(S' = \boldsymbol{(7, 0)}\)