Question

ar figures
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?
( (-4, 4) )
( (-2, 4) )
( (-4, 8) )
( (-2, 6) )

Explanation:

Step1: Find coordinates of N and P

From the graph, \( N=(0, 2) \), \( P=(2, -2) \)? Wait, no, looking at the grid, let's re - check. Wait, M is at (-1, -2)? No, wait the grid: Let's see, P is at (2, -2)? Wait no, the triangle: M is at (-1, -2), P is at (2, -2), N is at (0, 2). Wait, the center of dilation is P. Wait, the dilation rule is \( D_{P,2}(x,y) \), which means we dilate with center P and scale factor 2. The formula for dilation with center \( (a,b) \) and scale factor \( k \) is \( (x',y')=(a + k(x - a),b + k(y - b)) \)

First, find coordinates of N and P. From the graph: Let's assume the grid has each square as 1 unit. Let's find P: looking at the triangle, P is at (2, -2)? Wait no, M is at (-1, -2), P is at (2, -2), N is at (0, 2). Wait, center of dilation is P, so \( P=(2,-2) \), \( N=(0,2) \)

Step2: Apply dilation formula

The dilation formula with center \( (a,b) \) and scale factor \( k \) is \( (x',y')=(a + k(x - a),b + k(y - b)) \)

Here, \( a = 2 \), \( b=-2 \), \( k = 2 \), \( x = 0 \), \( y = 2 \)

First, calculate \( x - a=0 - 2=-2 \), \( y - b=2-(-2)=4 \)

Then, \( x'=2+2\times(-2)=2 - 4=-2 \)

\( y'=-2+2\times4=-2 + 8 = 6 \)

Wait, that gives (-2,6). Wait, maybe I misread P's coordinates. Let's re - check the graph. Wait, maybe P is at (2, -1)? No, let's look again. Wait, the original triangle: M is at (-1, -1), P is at (2, -1), N is at (0, 2). Wait, maybe my initial coordinate reading was wrong. Let's take another approach. The vector from P to N: \( \overrightarrow{PN}=N - P \). Let's find P and N correctly.

Looking at the graph, let's list the coordinates:

From the grid, let's assume:

• M: (-1, -2)
• P: (2, -2)
• N: (0, 2)

Center of dilation is P(2, -2), scale factor 2.

Vector \( \overrightarrow{PN}=N - P=(0 - 2,2-(-2))=(-2,4) \)

After dilation, the vector becomes \( 2\times\overrightarrow{PN}=(-4,8) \)

Then, \( N'=P + 2\times\overrightarrow{PN}=(2, -2)+(-4,8)=(2-4,-2 + 8)=(-2,6) \)

Answer:

(-2, 6) (corresponding to the option (-2,6))