Translating Points in the Plane, find the coordinates of the...

Translating Points in the Plane,
find the coordinates of the vertices of the polygon after
the indicated translation to a new position in the plane.
Original coordinates of vertices: $(-7,-2),(-2,2),$
$(-2,-4),(-7,-4)$
Shift: eight units up, four units to the right

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This allows geometric problems to be solved algebraically and provides a precise method to represent translations, rotations, and other transformations by manipulating the coordinates of points.

Vector Addition

Performing a translation in the coordinate plane can be understood as adding a translation vector to each point's coordinates. For example, if the translation vector is (h, k), then each point (x, y) is moved to (x + h, y + k). This method of vector addition is fundamental in applying translations across the plane.

Translation Transformation

A translation is a type of geometric transformation that moves every point of a shape or figure the same distance in a given direction. In the context of coordinate geometry, this means that a constant is added to the x-coordinate and/or the y-coordinate of every point, resulting in a new set of coordinates for the figure without altering its size, shape, or orientation.

Transcript

00:04 We are given the original coordinates of vertices of a polygon and the translation of this polygon to a new position in the plane, and we're asked to find the coordinates of the vertices of the polygon after this translation.

00:27 So the original coordinates of the vertices are negative 7, negative 2, negative 2, 2, 2, negative 4.

00:48 And negative 7, negative 4.

00:56 And our translation tells us to shift 8 units up and 4 units to the right.

01:15 Now, in order to shift 8 units up, we will add 8 to each y coordinate, and in order to shift 4 units to the right, we'll add 4 to each x coordinate.

01:33 So this results in the 4 new vertices with coordinates, negative 7 plus 4, negative 2 plus 8, negative 2 plus 4, 2 plus 8, negative 2 plus 4, negative 2 plus 8, and negative 7 plus 4, negative 4 plus 8.

02:10 Evaluating, we get the 4 vertices, negative 3, 6, 2, 10, 10.

02:27 2, 4, and negative 3, 4.

02:40 So this is our answer.

02:44 Now to check our answer, let's actually sketch a graph of these four points and see how the original 4 points, and sketch the transformation of these 4 points.

03:00 So to do this, notice that the x coordinates range from negative 2 to negative 7, and the y coordinates range from negative 2, to or negative 4 to positive 2.

03:13 So only draw the left side of the x -axis.

03:16 I'll draw the negative and positive y -axis.

03:25 Actually, i'll draw the positive x -axis 2, and we'll see why.

03:33 So the x -axis will range from negative 8 to positive 8, let's say.

03:43 In the y -axis, this will range from negative 4.

03:50 Let's range from negative 10 to positive 10.

04:06 Now, we have vertices at negative 7, negative 2, which is located about here, at negative 2, 2, 2, which is located about here, negative 2, negative 4, which is located around here, and negative 7, negative 4, just located about here...

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