2. Triangle GHI with vertices G(-1,6), H(-1,3), and I(-6, 6): a) translation along the vector (7,-5)

b) reflection in the line y = -3

a) The vertices of the triangle G'H'I' are , and , respectively.

b) The triangle G''H''I'' have vertices at , and .

Step-by-step explanation:

a) From Linear Algebra, we define the translation by the following vector:

(1)

Where:

- Original point, dimensionless.

- Translated point, dimensionless.

- Translation vector, dimensionless.

If we know that , , and , then the vertices of the triangle G'H'I' are, respectively:

(2)

(3)

(4)

The vertices of the triangle G'H'I' are , and , respectively.

b) From Linear Algebra, we define the reflection with respect to a horizontal line as follows:

(5)

Where:

- Original point, dimensionless.

- Reflection point, dimensionless.

If we know that and , then the location of point G'' is:

(6)

If we know that and , then the location of the point H'' is:

(7)

If we know that and , then the location of the point I'' is:

(8)

The triangle G''H''I'' have vertices at , and .

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step-by-step explanation: