Question

triangle mnp will be dilated according to the rule ( d_{p,2}(x,y) ), where point ( p ) is the center of dilation. what will be the coordinates of vertex ( n ) of the image? ( (-2,4) ) ( (-2,6) ) ( (-4,4) ) ( (-4,8) )

Explanation:

Step1: Find the vector from center of dilation \(P\) to \(N\)

Coordinates of \(P=(2, - 2)\), coordinates of \(N=(0,2)\).
The vector \(\overrightarrow{PN}=(0 - 2,2-(-2))=(-2,4)\)

Step2: Apply dilation rule

Dilation rule \(D_{P,2}(x,y)\) means multiply the vector \(\overrightarrow{PN}\) by scale factor \(2\).
New vector \(\overrightarrow{PN'} = 2\times(-2,4)=(-4,8)\)

Step3: Find coordinates of \(N'\)

Since \(P=(2,-2)\), and \(\overrightarrow{PN'}=(x_{N'}-2,y_{N'}+2)=(-4,8)\)
\(x_{N'}-2=-4\Rightarrow x_{N'}=-2\), \(y_{N'}+2 = 8\Rightarrow y_{N'}=6\)

Answer:

\((-2,6)\)