To describe the translation of a triangle on a coordinate plane where it is moved 4 units to the right and 3 units down, we need to determine how each coordinate point [tex]\((x, y)\)[/tex] changes as a result of the translation.
First, let's understand the effect of moving 4 units to the right. When a point on the coordinate plane is moved to the right, the [tex]\(x\)[/tex]-coordinate increases by the number of units moved. Thus, if we move a point [tex]\((x, y)\)[/tex] 4 units to the right, the new [tex]\(x\)[/tex]-coordinate will be [tex]\(x + 4\)[/tex].
Next, let's understand the effect of moving 3 units down. When a point on the coordinate plane is moved down, the [tex]\(y\)[/tex]-coordinate decreases by the number of units moved. Thus, if we move a point [tex]\((x, y)\)[/tex] 3 units down, the new [tex]\(y\)[/tex]-coordinate will be [tex]\(y - 3\)[/tex].
Combining these two transformations, the rule for the translation of a point [tex]\((x, y)\)[/tex] after moving it 4 units to the right and 3 units down can be written as:
[tex]\[
(x, y) \rightarrow (x + 4, y - 3)
\][/tex]
Now let's compare this rule with the given options:
1. [tex]\((x, y) \rightarrow (x + 3, y - 4)\)[/tex] — This rule suggests increasing the [tex]\(x\)[/tex]-coordinate by 3 and decreasing the [tex]\(y\)[/tex]-coordinate by 4, which does not match our translation.
2. [tex]\((x, y) \rightarrow (x + 3, y + 4)\)[/tex] — This rule suggests increasing the [tex]\(x\)[/tex]-coordinate by 3 and increasing the [tex]\(y\)[/tex]-coordinate by 4, which does not match our translation.
3. [tex]\((x, y) \rightarrow (x + 4, y - 3)\)[/tex] — This rule suggests increasing the [tex]\(x\)[/tex]-coordinate by 4 and decreasing the [tex]\(y\)[/tex]-coordinate by 3, which accurately describes our translation.
4. [tex]\((x, y) \rightarrow (x + 4, y + 3)\)[/tex] — This rule suggests increasing the [tex]\(x\)[/tex]-coordinate by 4 and increasing the [tex]\(y\)[/tex]-coordinate by 3, which does not match our translation.
Therefore, the rule that correctly describes the translation of the triangle is:
[tex]\[
(x, y) \rightarrow (x + 4, y - 3)
\][/tex]
So the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
