Question

5 from unit 6, lesson 1
reflect triangle abc over the line y = 2.
translate the image by the directed line segment from (0, 0) to (3, 2).
what are the coordinates of the vertices in the final image?
6 from unit 5, lesson 17
the density of water is 1 gram per cm³. an object floats in water if its density is less than waters. will a cylindrical log with its density is greater than waters sink or float?

Explanation:

Step1: Identify original coordinates

Original vertices: $A(1, 1)$, $B(-5, 3)$, $C(-2, 2)$

Step2: Reflect over $y=2$

For a point $(x,y)$, reflection over $y=k$ is $(x, 2k-y)$. Here $k=2$, so $2k-y=4-y$.

• $A'(1, 4-1)=(1, 3)$
• $B'(-5, 4-3)=(-5, 1)$
• $C'(-2, 4-2)=(-2, 2)$

Step3: Translate by $(3,2)$

Add $(3,2)$ to each reflected coordinate: $(x+3, y+2)$

• $A''(1+3, 3+2)=(4, 5)$
• $B''(-5+3, 1+2)=(-2, 3)$
• $C''(-2+3, 2+2)=(1, 4)$

Answer:

The coordinates of the final image vertices are $A''(4, 5)$, $B''(-2, 3)$, and $C''(1, 4)$