# Question Video: Translating a Triangle Using Congruent Triangles Mathematics • 11th Grade

Fill in the blank: In the figure, the triangles ππΏπ, πΏππ, πΏππ, and πππ are congruent. Then, the image of triangle πππ by translation of magnitude ππ in the direction of the ray ππ is οΌΏ.

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### Video Transcript

Fill in the blank. In the figure, the triangles ππΏπ, πΏππ, πΏππ, and πππ are congruent. Then, the image of triangle πππ by translation of magnitude ππ in the direction of the ray ππ is what.

In this question, we have been asked to find the image of the triangle πππ after a translation of magnitude ππ in the direction of the ray ππ. Letβs start by considering the four congruent triangles we have been given in the question. These are triangle ππΏπ, triangle πΏππ, triangle πΏππ, and triangle πππ.

Using the congruent triangles, we can label the congruent angles in the figure. We see that the three angles at π are three internal angles of any one of the triangles. Since the angles in a triangle sum to 180 degrees, the three angles at π must also sum to 180 degrees. Hence, πππ is a straight line segment. Similarly, the angles at π must sum to 180 degrees. So, πππ is also a straight line segment.

We have been asked to translate the triangle πππ. We can do this by translating each of the points π, π, and π to find the image of the triangle after the translation. Letβs start the translation by translating point π. We need to translate it with a magnitude of ππ and in the direction of the ray ππ. Since πππ is a straight line segment, this is the same as the direction of the ray ππ. Translating the point π in the direction of the ray ππ by a magnitude of ππ will move it to the point π. Hence, the image of the point π is the point π. Since the length of the line segment ππ and ππ are the same, we find that translating the point π with magnitude ππ or ππ in the direction of the ray ππ gives us that the image of point π is the point π.

Now all we need to do is translate the point π. Since πππ is a straight line segment and the angles πΏππ and πππ are equal, the direction of the ray ππ is equal to the direction of the ray ππΏ. Also, the length of ππ is equal to the length of πΏπ. Therefore, translating the point π with a magnitude of ππ in the direction of the ray ππ will move it to the point πΏ. Here, we reach our solution, which is that the image of triangle πππ after a translation of magnitude ππ in the direction of the ray ππ is the triangle ππΏπ.