Question

1. ( a(-10,-8) ) is the image of ( a ) after a dilation centered at the origin with a scale factor of 2. what are the coordinates of ( a )?
2. ( p(-2,5) ) is the image of ( p ) after a dilation centered at the origin with a scale factor of ( 1 / 3 ). what are the coordinates of ( p )?
3. ( z(3,8) ) is the image of ( z ) after a dilation centered at the origin with a scale factor of ( 3 / 2 ). what are the coordinates of ( z )?
4. ( h(1.5,-4) ) is the image of ( h ) after a dilation centered at the origin with a scale factor of ( 1 / 2 ). what are the coordinates of ( h )?

Explanation:

Step1: Formula de dilatación

Si un punto \(P(x,y)\) se dilata con centro en el origen y factor de escala \(k\), la imagen \(P'(x',y')\) se obtiene como \(x' = kx\) y \(y' = ky\). Entonces, para encontrar el punto original \(P(x,y)\) cuando se conoce \(P'(x',y')\) y \(k\), se utiliza \(x=\frac{x'}{k}\) y \(y = \frac{y'}{k}\).

Step2: Aplicar la fórmula para \(H'\)

Dado \(H'(1.5,-4)\) y \(k=\frac{1}{2}\).
Para la coordenada \(x\): \(x=\frac{1.5}{\frac{1}{2}}=1.5\times2 = 3\).
Para la coordenada \(y\): \(y=\frac{-4}{\frac{1}{2}}=-4\times2=-8\).

Answer:

Las coordenadas de \(H\) son \((3,-8)\)