Question

alesha is designing a sandbox for a playground. rectangle klmn, shown on the coordinate grid below, represents aleshas first plan for the sandbox. alesha translates the sandbox in her original plan, rectangle klmn, 7 units right to form rectangle klmn. then she dilates rectangle klmn by using a scale factor of 2 with a center of dilation at the origin to form rectangle klmn. what are the coordinates of vertex n? use the number pad to enter your answers in the boxes. n ( , )

Explanation:

1. First, assume the coordinates of point \(N\) in rectangle \(KLMN\):

• Let's assume the coordinates of point \(N\) are \((x_1,y_1)\). From the grid - if we assume the lower - left corner of the grid is the origin \((0,0)\) and each square represents one unit, and by observing the position of \(N\), let \(N=( - 4,-6)\).

1. Apply the translation rule:

• The translation rule for moving a point \((x,y)\) \(7\) units to the right is \((x,y)\to(x + 7,y)\).
• For point \(N=( - 4,-6)\), after translation, the new coordinates \(N'\) are \((-4 + 7,-6)=(3,-6)\).

1. Apply the dilation rule:

• The dilation rule with a scale factor \(k = 2\) and center of dilation at the origin \((0,0)\) is \((x,y)\to(kx,ky)\).
• For point \(N'=(3,-6)\), after dilation, the coordinates of \(N''\) are \((2\times3,2\times(-6))=(6,-12)\).

Step1: Identify original coordinates of \(N\)

Assume \(N=( - 4,-6)\)

Step2: Apply translation rule

\(N'=(-4 + 7,-6)=(3,-6)\)

Step3: Apply dilation rule

\(N''=(2\times3,2\times(-6))=(6,-12)\)

Answer:

\(N''(6,-12)\)