Question

date: this is a 2 - page document!
directions: give each rule for counterclockwise rotations about the origin:
90°: (x,y)→ 180°: (x,y)→ 270°: (x,y)→
directions: graph and label each figure and its image under a rotation about the origin.
give the coordinates of the image.

1. rhombus abcd with vertices a(2,6), b(6,7), c(5,3), and d(1,2): 180°

a (__,__)
b (__,__)
c (__,__)
d (__,__)

1. trapezoid stuv with vertices s(-7,-1), t(- u(-2,-5), and v(-7,-7): 90° counterclocky

s (____
t (____
u (____
v (____

1. triangle fgh with vertices f(-7,8), g(-1,1), and h(-8,4): 270° counterclockwise

f (__,__)
g (__,__)
h (__,__)

1. square jklm with vertices j(1,-3) l(8,-4), and m(4,-7): 90° counte
2. quadrilateral wxyz with vertices w(-6,7), x(-3,6), y(-1,3), and z(-7,1): 180°

w (__,__)
x (__,__)
y (__,__)

1. rectangle cdef with verti e(6,5), and f(5,3): 270°

Explanation:

Step1: Determine rotation rules

• For a \(90^{\circ}\) counter - clockwise rotation about the origin, the rule is \((x,y)\to(-y,x)\).
• For a \(180^{\circ}\) rotation about the origin, the rule is \((x,y)\to(-x,-y)\).
• For a \(270^{\circ}\) counter - clockwise rotation about the origin, the rule is \((x,y)\to(y, - x)\).

Step2: Apply \(180^{\circ}\) rule to rhombus \(ABCD\)

• Given \(A(2,6)\), using \((x,y)\to(-x,-y)\), we get \(A'(- 2,-6)\).
• Given \(B(6,7)\), using \((x,y)\to(-x,-y)\), we get \(B'(-6,-7)\).
• Given \(C(5,3)\), using \((x,y)\to(-x,-y)\), we get \(C'(-5,-3)\).
• Given \(D(1,2)\), using \((x,y)\to(-x,-y)\), we get \(D'(-1,-2)\).

Step3: Apply \(90^{\circ}\) rule to trapezoid \(STUV\) (assuming \(T(-4, - 3)\) as vertex \(T\) value is cut - off in the problem, general rule application)

• For \(S(-7,-1)\), using \((x,y)\to(-y,x)\), \(S'(1,-7)\).
• For \(T(-4,-3)\) (assuming), using \((x,y)\to(-y,x)\), \(T'(3,-4)\).
• For \(U(-2,-5)\), using \((x,y)\to(-y,x)\), \(U'(5,-2)\).
• For \(V(-7,-7)\), using \((x,y)\to(-y,x)\), \(V'(7,-7)\).

Step4: Apply \(270^{\circ}\) rule to triangle \(FGH\)

• For \(F(-7,8)\), using \((x,y)\to(y,-x)\), \(F'(8,7)\).
• For \(G(-1,1)\), using \((x,y)\to(y,-x)\), \(G'(1,1)\).
• For \(H(-8,4)\), using \((x,y)\to(y,-x)\), \(H'(4,8)\).

Step5: Apply \(180^{\circ}\) rule to quadrilateral \(WXYZ\)

• For \(W(-6,7)\), using \((x,y)\to(-x,-y)\), \(W'(6,-7)\).
• For \(X(-3,6)\), using \((x,y)\to(-x,-y)\), \(X'(3,-6)\).
• For \(Y(-1,3)\), using \((x,y)\to(-x,-y)\), \(Y'(1,-3)\).
• For \(Z(-7,1)\), using \((x,y)\to(-x,-y)\), \(Z'(7,-1)\).

Answer:

• \(90^{\circ}:(x,y)\to(-y,x)\); \(180^{\circ}:(x,y)\to(-x,-y)\); \(270^{\circ}:(x,y)\to(y,-x)\)
• \(A'(-2,-6)\), \(B'(-6,-7)\), \(C'(-5,-3)\), \(D'(-1,-2)\)
• \(S'(1,-7)\), \(T'(3,-4)\) (assuming \(T(-4,-3)\)), \(U'(5,-2)\), \(V'(7,-7)\)
• \(F'(8,7)\), \(G'(1,1)\), \(H'(4,8)\)
• \(W'(6,-7)\), \(X'(3,-6)\), \(Y'(1,-3)\), \(Z'(7,-1)\)