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?

Explanation:

Step1: Translate point P in x - direction for part (a)

When we translate 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 $P=(a,b)$, after translating $P$ by $c$ units in the $x$-direction, the coordinates of $Q$ are $(a + c,b)$.

Step2: Translate point Q in y - direction for part (a)

When we translate 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 $Q=(a + c,b)$, the coordinates of $R$ are $(a + c,b + d)$.

Step3: Translate point P in y - direction for part (b)

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

Step4: Translate point S in x - direction for part (b)

When we translate 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. Since $S=(a,b + d)$, the coordinates of $T$ are $(a + c,b + d)$.

Answer:

(a) $(a + c,b + d)$
(b) $(a + c,b + d)$