Question

determine whether the statement is always, sometimes, or never true. justify your argument.
a translation along < a, b > followed by the translation along < c, d > is the translation along < a + c, b + d >
always: the first translation maps (x, y) to (x+ a, y+ b), and the second translation maps this image to (x+
a + c, y+ b + d), which is not equivalent to the translation along < a + c, b + d >.

Explanation:

Step1: Analyze the first translation

The first translation along \(\langle a,b
angle\) maps a point \((x,y)\) to \((x + a,y + b)\) according to the rule of translation in the coordinate plane.

Step2: Analyze the second translation

The second translation along \(\langle c,d
angle\) maps the point \((x + a,y + b)\) (the result of the first translation) to \((x+a + c,y + b + d)\) using the translation rule.

Step3: Compare with the combined - translation rule

A single translation along \(\langle a + c,b + d
angle\) maps the point \((x,y)\) to \((x+(a + c),y+(b + d))=(x + a + c,y + b + d)\)

Answer:

The statement is always true.