Triangle ABC with vertices A(0, 2), B(-3, -1), and C(-2, -4) is translated 1 unit right and 3 un

Triangle ABC with vertices A(0, 2), B(-3, -1), and C(-2, -4)...

Key Concepts

Graphing in the Coordinate Plane

Graphing in the coordinate plane involves plotting the points of a figure on a grid defined by the x- and y-axes. This visual representation is important for verifying transformations like translations, as it helps in comparing the original position (preimage) with the new position (image) of the figure.

Translation

Translation is a type of geometric transformation that moves every point of a figure by the same distance in the same direction. In coordinate geometry, this is done by adding a constant value to the x-coordinate and/or the y-coordinate of every point, making it a rigid motion that maintains the size and shape of the figure.

Coordinate Transformation

Coordinate transformation involves the operation of changing the coordinates of points in a figure according to a rule or function. For translations, the transformation rule is applied uniformly across all points, leading to a new set of coordinates that represent the shifted figure.

Preimage and Image

In geometric transformations, the preimage is the original figure before any transformation is applied, whereas the image is the figure after the transformation. Understanding these terms is essential since it distinguishes the state of a figure before and after the transformation process.

Transcript

00:01 Okay, so in this problem, this is kind of an extension of the problem right before this one, where you have been given three vertices of a triangle, and they want you to translate it.

00:14 So if we write the vertex matrix, remember the x is, we put on top, and the y is we put on the bottom of the vertex matrix.

00:23 And then i know that i am moving this three units, one unit to the right.

00:29 So remember the x, y, x, if you go to the right, you're positive.

00:35 If you go up on the y -axis, it's also positive.

00:39 So this is going to be adding one to the x's and then also adding three to that.

00:48 So we add these together.

00:51 And so my translation matrix becomes zero plus 1, negative 3, plus 1, negative 2, plus 1.

01:00 2 plus 3, negative 1 plus 3, and negative 4 plus 3.

01:08 Okay, and so this translation matrix, 0 plus 1 is 1, negative 3 plus 1 is negative 2.

01:17 Negative 2 plus 1 is negative 1, 2 plus 1, 2 plus 3 is 5, negative 1 plus 3 is positive 2, and negative 4 plus 3 is negative 1.

01:26 So that's my translation matrix, which then we can write our new resulting vertices.

01:35 We would call those a prime would be 1 -5.

01:39 B -prime would be negative 2, 2, and then c -prime would be negative 1 -7.

01:49 So we have our new vertices.

01:51 So let's just take a minute and think if we wanted to graph these.

01:55 Okay if we wanted to graph the original ones try to draw this as best as i can to scale here it's not going to be perfect sorry one two three four okay so our original one our original a was at zero two so x is zero put a point here zero two there make a big um b so this was a b was at negative 3 3 negative 1.

02:35 So let's be there...

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