Question

1. let p be a point with coordinates (a,b), and assume that c and d are positive numbers. (a) translate the point p by c units in the x - direction to obtain a point q, then translate q by d units in the y - direction to obtain the point r. what are the coordinates of the point r? (b) translate the point p by d units in the y - direction to obtain a point s, then translate s by c units in the x - direction to obtain a point t. what are the coordinates of the point t? (c) compare your answers for parts (a) and (b). what have you demonstrated? (answer in complete sentences.)

Explanation:

Step1: Translate point P to get Q

When translating a point $(x,y)$ by $c$ units in the $x -$direction, the new $x$-coordinate is $x + c$ and the $y$-coordinate remains the same. Given $P=(a,b)$, when we translate $P$ by $c$ units in the $x$-direction to get $Q$, the coordinates of $Q$ are $(a + c,b)$.

Step2: Translate Q to get R

When translating a point $(x,y)$ by $d$ units in the $y -$direction, the $x$-coordinate remains the same and the new $y$-coordinate is $y + d$. Since $Q=(a + c,b)$, when we translate $Q$ by $d$ units in the $y$-direction to get $R$, the coordinates of $R$ are $(a + c,b + d)$.

Step3: Translate point P to get S for part (b)

When translating $P=(a,b)$ by $d$ units in the $y$-direction to get $S$, the coordinates of $S$ are $(a,b + d)$.

Step4: Translate S to get T

When translating $S=(a,b + d)$ by $c$ units in the $x$-direction to get $T$, the coordinates of $T$ are $(a + c,b + d)$.

Step5: Compare answers

The coordinates of $R$ from part (a) and $T$ from part (b) are the same, $(a + c,b + d)$. This demonstrates that translations in the $x$ and $y$ directions are commutative, meaning the order in which we perform translations in the $x$ and $y$ directions does not affect the final position of the point.

Answer:

(a) $(a + c,b + d)$
(b) $(a + c,b + d)$
(c) Translations in the $x$ and $y$ directions are commutative.