Question

lesson 3 - session 2 performing sequences of rigid tran > perform the given sequence of transformations on each figure coordinates of the vertices of the final image. then tell whethe is congruent to the original figure. 1 reflect across the x - axis. translate 5 units left. 2 rotate 90 reflect a 3 translate 2 units right and 4 units down. rotate 180° around the origin. 4 reflec coun

Explanation:

1. For the triangle \(ABC\) with vertices \(A(3,6)\), \(B(5,2)\), \(C(2,2)\):

• Step 1: Reflection across the \(x -\)axis
• The rule for reflecting a point \((x,y)\) across the \(x -\)axis is \((x,-y)\).
• For \(A(3,6)\), the new - point \(A_1=(3, - 6)\); for \(B(5,2)\), the new - point \(B_1=(5,-2)\); for \(C(2,2)\), the new - point \(C_1=(2,-2)\).
• Step 2: Translation 5 units left
• The rule for translating a point \((x,y)\) 5 units left is \((x - 5,y)\).
• For \(A_1(3,-6)\), the new - point \(A_2=(3 - 5,-6)=(-2,-6)\); for \(B_1(5,-2)\), the new - point \(B_2=(5 - 5,-2)=(0,-2)\); for \(C_1(2,-2)\), the new - point \(C_2=(2 - 5,-2)=(-3,-2)\).
• Since reflection and translation are rigid motions, the final image \(\triangle A_2B_2C_2\) is congruent to the original \(\triangle ABC\).

1. For the quadrilateral \(HIJK\) with vertices \(H(-6,5)\), \(I(-2,5)\), \(J(-2,2)\), \(K(-4,2)\):

• Step 1: Translation 2 units right and 4 units down
• The rule for translating a point \((x,y)\) 2 units right and 4 units down is \((x + 2,y-4)\).
• For \(H(-6,5)\), the new - point \(H_1=(-6 + 2,5 - 4)=(-4,1)\); for \(I(-2,5)\), the new - point \(I_1=(-2 + 2,5 - 4)=(0,1)\); for \(J(-2,2)\), the new - point \(J_1=(-2 + 2,2 - 4)=(0,-2)\); for \(K(-4,2)\), the new - point \(K_1=(-4 + 2,2 - 4)=(-2,-2)\).
• Step 2: Rotation 180° around the origin
• The rule for rotating a point \((x,y)\) 180° around the origin is \((-x,-y)\).
• For \(H_1(-4,1)\), the new - point \(H_2=(4,-1)\); for \(I_1(0,1)\), the new - point \(I_2=(0,-1)\); for \(J_1(0,-2)\), the new - point \(J_2=(0,2)\); for \(K_1(-2,-2)\), the new - point \(K_2=(2,2)\).
• Since translation and rotation are rigid motions, the final image of the quadrilateral is congruent to the original quadrilateral.

Answer:

1. For \(\triangle ABC\): The vertices of the final image are \(A_2(-2,-6)\), \(B_2(0,-2)\), \(C_2(-3,-2)\), and it is congruent to the original figure.
2. For quadrilateral \(HIJK\): The vertices of the final image are \(H_2(4,-1)\), \(I_2(0,-1)\), \(J_2(0,2)\), \(K_2(2,2)\), and it is congruent to the original figure.