Question

write the composition of transformations as one transformation.

1. $t_{(4,5)}circ t_{(3,1)}$
2. $t_{(-1,-3)}circ t_{(2,-2)}$
3. $t_{(1,1)}circ t_{(-4,-5)}$

given $\triangle xyz$ with vertices $x(-2,1), y(-1,3),$ and $z(-4,2)$, write the translation equivalent to the composition of transformations. suppose the equation of line $m$ is $x = 5$, the equation of line $n$ is $y = 4$, and the equation of line $p$ is $x = 3$.

1. $r_{m}circ r_{y - axis}$
2. $r_{n}circ r_{x - axis}$
3. $r_{p}circ r_{y - axis}$
4. understand how far apart are two parallel lines $l$ and $m$ such that $t_{(4,0)}(\triangle def)=(r_{m}circ r_{l})(\triangle def)$?
5. apply suppose the streets of a city form a grid. the composition of rigid motions $t_{(a,b)}circ t_{(c,d)}$ describes the route of a limousine through the city from its starting position. how would you describe the route in words?

Explanation:

Step1: Recall translation composition rule

For two translations $T_{(a,b)}$ and $T_{(c,d)}$, the composition $T_{(a,b)}\circ T_{(c,d)}=T_{(a + c,b + d)}$.

Step2: Solve problem 6

For $T_{(4,5)}\circ T_{(3,1)}$, we have $a = 4,b = 5,c = 3,d = 1$. Then $a + c=4 + 3=7$ and $b + d=5 + 1=6$. So $T_{(4,5)}\circ T_{(3,1)}=T_{(7,6)}$.

Step3: Solve problem 7

For $T_{(-1,-3)}\circ T_{(2,-2)}$, we have $a=-1,b = - 3,c = 2,d=-2$. Then $a + c=-1+2 = 1$ and $b + d=-3+( - 2)=-5$. So $T_{(-1,-3)}\circ T_{(2,-2)}=T_{(1,-5)}$.

Step4: Solve problem 8

For $T_{(1,1)}\circ T_{(-4,-5)}$, we have $a = 1,b = 1,c=-4,d=-5$. Then $a + c=1+( - 4)=-3$ and $b + d=1+( - 5)=-4$. So $T_{(1,1)}\circ T_{(-4,-5)}=T_{(-3,-4)}$.

Answer:

1. $T_{(7,6)}$
2. $T_{(1,-5)}$
3. $T_{(-3,-4)}$