Question

Write the rule for finding the coordinates of a point that has undergone a dilation (with the origin as the point of dilation). Use this rule to find the coordinates of this point’s dilation from the origin, with a scale factor of : (4, 10).

Answer 1

The rule for dilation (scale factor k) about the origin is given by:
(x,y)->(kx,ky)
For the point P(4,10)
the image is therefore
P(4,10)->P'(4k,10k) [ since k is not numerically given ]
When k is known numerically, substitute k into P' to find the numerical coordinates.

Answer 2

Point (4,10) goes by a dilation of factor m or 1/2 ( With origin as a point of Dilation)=[4 m, 10 m] [4 × 1/2, 10× 1/2]⇒[2,5].

Explanation:

Suppose a point has coordinate (x,y).

1.The meaning of Dilation of a point is moving to a new point which is at larger distance from origin if we are Dilating by a value i.e an integer in Coordinate System

2. Or If we are Dilating by a fraction which lies between 0<x<1, then distance from origin decreases.

Now , the point (x,y) goes through a Dilation by a factor m(with the origin as point of dilation) , Then New Coordinate of the point= [ m x, my]

Earlier , Distance from origin=x² + y²

Now,Distance from Origin= m² [x² + y²]

If point (4,10) goes through a dilation by factor m ( with the origin as point of dilation)

=(4 m ,10 m)