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?
$\bigcirc$ $(-2, 4)$
$\bigcirc$ $(-2, 6)$
$\bigcirc$ $(-4, 4)$
$\bigcirc$ $(-4, 8)$

Explanation:

Step1: Find coordinates of N and P

From the graph, \( N(0, 2) \) and \( P(2, -2) \) (assuming grid units, P is at (2, -2) as per the triangle's base). Wait, actually, looking at the grid: N is at (0, 2), M is at (-1, -2), P is at (2, -2)? Wait, no, the triangle has M, N, P. Let's recheck: N is at (0, 2), M at (-1, -2), P at (2, -2)? Wait, maybe better to get vector from P to N.

Wait, dilation with center P, scale factor 2. So the rule for dilation with center \( (h,k) \) and scale factor \( s \) is \( (x', y') = (h + s(x - h), k + s(y - k)) \).

First, find coordinates of N and P. From the graph: N is at (0, 2), P is at (2, -2)? Wait, no, the triangle's base: M is at (-1, -2), P is at (2, -2), so P is (2, -2), N is (0, 2).

So vector from P to N: \( (0 - 2, 2 - (-2)) = (-2, 4) \). Dilation by scale factor 2: new vector is \( 2*(-2, 4) = (-4, 8) \). Then add to P's coordinates: \( (2 + (-4), -2 + 8) = (-2, 6) \). Wait, let's do it step by step.

Step1: Identify coordinates

From the grid:

• Point P: Let's see, the base of the triangle is from M (-1, -2) to P (2, -2), so P is (2, -2).
• Point N: (0, 2).

Step2: Apply dilation rule

Dilation with center P (h=2, k=-2) and scale factor 2. The formula for dilation with center \( (h,k) \) is:
\( (x', y') = (h + 2(x - h), k + 2(y - k)) \)
Substitute x=0, y=2, h=2, k=-2:
\( x' = 2 + 2(0 - 2) = 2 - 4 = -2 \)
\( y' = -2 + 2(2 - (-2)) = -2 + 2(4) = -2 + 8 = 6 \)
So N' is (-2, 6).

Answer:

(-2, 6) (Option B: (-2, 6))