Find the vertices of the image of triangle PQR having the vertices P(-1,2) Q(-3,5) and R(0,4) under the enlargement with scale factor -2 and centre at origin?

1 Answer
Apr 7, 2018

see explanation

Explanation:

Solution 1 :)
Let k be the scale factor, when k is negative, the "image" triangle is on the opposite side of the center of the dilation O (vertices are on opposite side of O from the preimage). Also note that the triangle has been rotated 180º.
enter image source here
In our case, k=-2, and the center of dilation is at the origin O, we can describe this transformation as a dilation of scale factor k=+2 combined with a rotation of 180^@.
1) Under a dilation centered at the origin with a scale factor (k>0),
a point A(x,y) -> A_1(kx,ky)
see the pre-image and image-1 as shown in the figure above,
=> P(-1,2) -> P_1(2xx(-1), " "2xx2)=color(red)(P_1(-2,4))
=> Q(-3,5) -> Q_1(2xx(-3), " "2xx5)=color(red)(Q_1(-6,10))
=> R(0,4) -> R_1(2xx0, " "2xx4)=color(red)(R_1(0,8)
2) Under a rotation of 180^@ about the origin,
a point A_1(kx,ky) -> A_2(-kx,-ky)
see image-2 as shown in the figure,
=> color(red)(P_1(-2,4)) -> color(blue)(P_2(2,-4))
=> color(red)(Q_1(-6,10)) -> color(blue)(Q_2(6,-10))
=> color(red)(R_1(0,8) -> color(blue)(R_2(0,-8)

solution 2 :)
enter image source here
note that when k < -1 , the image is larger, with a 180^@ rotation. The negative symbol indicates direction.
Formula for an enlargement with a negative scale factor (k < -1) and center at origin,
a point A(x,y) = A'(-kx, -ky)
=> P(-1,2) -> P'(-2xx(-1), -2xx2)=color(blue)(P'(2,-4)),
=> Q(-3,5) -> Q'(-2xx(-3), -2xx5)=color(blue)(Q'(6,-10)),
=> R(0,4) -> P'(-2xx(0), -2xx4)=color(blue)(R'(0,-8))