The transformation (D_(14)∘ r_((180^(∘),O)))(△RST) can be written as D_(14)( r_((180^(∘),O))(△RST)). This means that △JKL will be rotated 180^(∘) about the origin O and then the image will be dilated by the scale factor of 14.
R''(3/4,- 1/2), S''(1/4,- 1/4), T''(1/2,- 1)
Practice makes perfect
The transformation (D_(14)∘ r_((180^(∘),O)))(△RST) can be written as D_(14)( r_((180^(∘),O))(△RST)). This means that we will rotate △RST 180^(∘) about the origin O and then dilate the image by a scale factor of 14.
We will do these two things one at a time.
Rotation
Let's start by rotating △RST 180^(∘) about the origin. Unless we are told otherwise, we perform rotations counterclockwise. We will label the new image △R'S'T'.
Dilation
Let's dilate △R'S'T' by a scale factor of 14 and center of dilation O(0,0). To do so, we will find the images of the vertices by multiplying their coordinates by the scale factor. Let R'', S'', and T'' be their corresponding images. Therefore, D_(14) ( △R'S'T')= △R''S''T''.
△R'S'T'
△R''S''T''
R'(3,- 2)
R''(3* 14, - 2* 14) ⇒ R''( 34,- 12)
S'(1,- 1)
S''(1 * 14, - 1 * 14) ⇒ S''( 14,- 14)
T'(2,- 4)
T''(2* 14,- 4* 14) ⇒ T''( 12,- 1)
Since we need to find only coordinates of (D_(14)∘ r_((180^(∘),O)))(△RST), we do not have to graph a dilation. The coordinates of △RST after given trasnformatios are R''( 34,- 12), S''( 14,- 14), and T''( 12,- 1).
