1. **State the problem:** We have triangle $\triangle JKL$ with vertices $J(2,6)$, $K(1,4)$, and $L(4,4)$. We need to: - a) Translate $\triangle JKL$ by vector $[4,-2]$ to get $\triangle J'K'L'$. - b) Translate $\triangle J'K'L'$ by vector $[-3,-2]$ to get $\triangle J''K''L''$. - c) Find the single translation vector that maps $\triangle JKL$ directly onto $\triangle J''K''L''$. 2. **Formula for translation:** If a point $(x,y)$ is translated by vector $[a,b]$, the new point is: $$ (x', y') = (x + a, y + b) $$ 3. **Step a: Translate $\triangle JKL$ by $[4,-2]$** - $J'(x) = 2 + 4 = 6$, $J'(y) = 6 - 2 = 4$ so $J'(6,4)$ - $K'(x) = 1 + 4 = 5$, $K'(y) = 4 - 2 = 2$ so $K'(5,2)$ - $L'(x) = 4 + 4 = 8$, $L'(y) = 4 - 2 = 2$ so $L'(8,2)$ 4. **Step b: Translate $\triangle J'K'L'$ by $[-3,-2]$** - $J''(x) = 6 - 3 = 3$, $J''(y) = 4 - 2 = 2$ so $J''(3,2)$ - $K''(x) = 5 - 3 = 2$, $K''(y) = 2 - 2 = 0$ so $K''(2,0)$ - $L''(x) = 8 - 3 = 5$, $L''(y) = 2 - 2 = 0$ so $L''(5,0)$ 5. **Step c: Find the single translation vector from $\triangle JKL$ to $\triangle J''K''L''$** Calculate the vector from $J$ to $J''$: $$ (3 - 2, 2 - 6) = (1, -4) $$ Check with $K$ to $K''$: $$ (2 - 1, 0 - 4) = (1, -4) $$ Check with $L$ to $L''$: $$ (5 - 4, 0 - 4) = (1, -4) $$ All match, so the single translation vector is $[1, -4]$. **Final answers:** - $\triangle J'K'L'$ vertices: $J'(6,4), K'(5,2), L'(8,2)$ - $\triangle J''K''L''$ vertices: $J''(3,2), K''(2,0), L''(5,0)$ - Single translation vector mapping $\triangle JKL$ to $\triangle J''K''L''$: $[1, -4]$