Question

for exercises 8 - 10, use the graph to answer the questions. the vertices for a triangle are x(3, 1), y(2, - 5), and z(7, 0). the triangle will be translated 3 units up and 2 units to the left. 8. what are the coordinates of x, y, and z? 9. how does the length of xy compare to the length of xy? 10. how does the area of triangle xyz compare to the area of triangle xyz? for exercises 11 and 12, use the graph to answer the questions. 11. the image of quadrilateral abcd after being translated up 6 units and to the right 1 unit, abcd is shown. what were the original coordinates of a, b, c, and d? 12. how are the x - values of each coordinate changed under this translation? the y - values? reflect how can you explain to a classmate what a translation is? math is... mindset what did your classmates learn about your math journey?

Explanation:

Step1: Find new coordinates for triangle translation

For a translation 3 units up (add 3 to y - coordinate) and 2 units to the left (subtract 2 from x - coordinate).
For point $X(3,1)$: $X'=(3 - 2,1 + 3)=(1,4)$
For point $Y(2,-5)$: $Y'=(2 - 2,-5 + 3)=(0,-2)$
For point $Z(7,0)$: $Z'=(7 - 2,0 + 3)=(5,3)$

Step2: Analyze length comparison in translation

Translation is a rigid - motion. Rigid motions preserve distance. So, the length of $X'Y'$ is equal to the length of $XY$.

Step3: Analyze area comparison in translation

Translation is a rigid - motion. Rigid motions preserve area. So, the area of triangle $XYZ$ is equal to the area of triangle $X'Y'Z'$.

Step4: Find original coordinates of quadrilateral

For a translation up 6 units and right 1 unit, to find the original coordinates, subtract 1 from the x - coordinate and subtract 6 from the y - coordinate of the translated points. But since the coordinates of $A',B',C',D'$ are not given in the text (only the graph is shown), assume $A'(x_1,y_1),B'(x_2,y_2),C'(x_3,y_3),D'(x_4,y_4)$. Then $A=(x_1 - 1,y_1 - 6),B=(x_2 - 1,y_2 - 6),C=(x_3 - 1,y_3 - 6),D=(x_4 - 1,y_4 - 6)$

Step5: Analyze coordinate - value changes

For the translation up 6 units and right 1 unit, the x - values of each coordinate increase by 1, and the y - values of each coordinate increase by 6.

Answer:

1. $X'(1,4),Y'(0,-2),Z'(5,3)$
2. The length of $X'Y'$ is equal to the length of $XY$.
3. The area of triangle $XYZ$ is equal to the area of triangle $X'Y'Z'$.
4. Let the coordinates of $A',B',C',D'$ be $(x_{A'},y_{A'}),(x_{B'},y_{B'}),(x_{C'},y_{C'}),(x_{D'},y_{D'})$ respectively. Then $A=(x_{A'}-1,y_{A'}-6),B=(x_{B'}-1,y_{B'}-6),C=(x_{C'}-1,y_{C'}-6),D=(x_{D'}-1,y_{D'}-6)$
5. The x - values increase by 1, and the y - values increase by 6.