GEOMETRY PLEASEE FASTT

Question

Graph △DOG with vertices D(−2, 1), O(−4, −1), G(−1, −4)

Rotate the figure 270°about the origin, then dilate by a scale factor of 2 about the point (−6, 5)

Answer 1

This is a great question about 2D linear transformations on a triangle.

Step 1: Rotate △DOG About the Origin

To rotate a point 270° about the origin, we can use the following formula. P is the original point, and P' is the transformed point.

P = (x, y) → P' = (y, -x)

Let's do this to points D, O, and G.

D = (−2, 1) becomes D′ = (1, 2)

O = (−4, −1) becomes O′ = (−1, 4)

G = (−1, −4) becomes G′ = (−4, 1)

So after rotation, the triangle has vertices D′ = (1, 2), O′ = (−1, 4), G′ = (−4, 1).

Step 2: Dilate △DOG About a Reference Point by a Scale Factor:

To dilate a point about a reference point, we can use the following formula. 'k' is the scale factor and (a, b) are the coordinates of the reference point.

P = (x, y) → P' = (a + k(x − a), b + k(y − b))

We are told that the dilation factor is 2 and the reference point is (−6, 5). We can make the following definitions.

k = 2, a = -6, and b = 5

Let's do this to points D', O', and G'.

D′ = (1, 2) → D′′ = (−6 + 2(1 + 6), 5 + 2(2 − 5)) = (8, −1)

O′ = (−1, 4) → O′′ = (−6 + 2(−1 + 6), 5 + 2(4 − 5)) = (4, 3)

G′ = (−4, 1) → G′′ = (−6 + 2(−4 + 6), 5 + 2(1 − 5)) = (−2, −3)

After the rotation and dilation, the triangle’s vertices are D′′(8, −1), O′′(4, 3), and G′′(−2, −3).