Question

Map My Route

Zari and Taye are training for track season. They each go on training runs from their homes. Zari's house is
located at point A and Taye's house at point A' on the coordinate plane shown. Each unit on the coordinate
plane corresponds to one mile.
-14-12-10-8-6-4-
A'
14-
12-
10-
-8-
-6-
4-
2
2
2
4
6
8
10-
12
14-
2
A
4
6
8 10 12 14-
Ax
1. On one day of training, Zari runs 2 miles north, then 4 miles east, and then back to her house via the
shortest route. Taye runs 2 miles west, then 4 miles south, and then back to his house, again via the
shortest route.
Draw their routes on the coordinate plane. Demonstrate that the two triangles formed by their routes are
congruent. Include a triangle congruency statement in your response.
On another day of training, Zari runs on a straight path to a park located at (8, 12) on the grid, and from
there she runs on a straight path to the school located at (10, 6) on the grid. From the school, she runs on a
straight path directly back to her house. Taye runs on a straight path to a library located at (-12, 0) on the
grid, and from there he runs on a straight path to a tennis court located at (-10, 6) on the grid. From the
tennis court, he runs on a straight path directly back to his house.
Draw their routes on the coordinate plane. Demonstrate that the two triangles formed by their routes are
congruent. Include a triangle congruency statement in your response.

Answer 1

To show that the triangles formed by Zari and Taye's training routes are congruent, we analyze the coordinates of their routes. By comparing the endpoints of their routes, we can establish congruency using the SSS or SAS congruence criteria. The triangles formed are: triangle ABC is congruent to triangle A'B'C', and triangle XYZ is congruent to triangle X'Y'Z'.

To show that the two triangles formed by Zari and Taye's training routes are congruent, we need to prove that they have the same side lengths and angles. Let's analyze Zari's route first:

From (2, 2), Zari runs 2 miles north, which takes her to the point (2, 4).

Then, she runs 4 miles east, which takes her to the point (6, 4).

Finally, she returns to her house via the shortest route, which is 4 miles west. This takes her back to the point (2, 4), completing her route.

Now let's analyze Taye's route:

From (-6, -8), Taye runs 2 miles west, which takes him to the point (-8, -8).

Then, he runs 4 miles south, which takes him to the point (-8, -12).

Finally, he returns to his house via the shortest route, which is 4 miles north. This takes him back to the point (-8, -8), completing his route.

By comparing the coordinates of the endpoints of their routes, we can see that the two triangles formed by Zari and Taye's routes are congruent. We can write the triangle congruency statement as follows: by SSS congruence, triangle ABC is congruent to triangle A'B'C', where A = (2, 2), B = (6, 4), C = (2, 4), A' = (-6, -8), B' = (-8, -8), and C' = (-8, -12).

In the second training session, Zari's route can be represented as a straight line from (8, 12) to (10, 6), while Taye's route can be represented as a straight line from (-12, 0) to (-10, 6). By comparing the coordinates of the endpoints of their routes, we can see that the two triangles formed by Zari and Taye's routes are congruent. We can write the triangle congruency statement as follows: by SAS congruence, triangle XYZ is congruent to triangle X'Y'Z', where X = (8, 12), Y = (10, 6), Z = (2, 2), X' = (-12, 0), Y' = (-10, 6), and Z' = (-8, -8).