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: $(5,8),(3,6),(7,6)$
Shift: 6 units down, 10 units to the left

Key Concepts

Translation

A translation in coordinate geometry is a type of transformation that shifts every point in a figure by the same fixed distance in a specific direction, preserving the shape and orientation of the figure.

Translation Vector

The translation vector indicates the magnitude and direction of the shift. It is typically represented as (dx, dy), where dx is the horizontal shift and dy is the vertical shift.

Coordinate Transformation

Coordinate transformation involves updating the coordinates of points in a figure by applying the same translation vector to each point, thereby obtaining their new positions in the plane.

Transcript

00:03 We are given the original coordinates of the vertices of a polygon and a translation on this polygon to a new position in the plane, and we're asked to find the coordinates of the vertices of the new polygon after this translation.

00:22 So the original coordinates of the vertices are 5 -8, 3, 6, and 7 -6, and the shift is 6 units down, and 10 units.

00:36 To the left.

00:39 First, i'm going to graph the three vertices of this polygon in order to sketch the polygon.

00:46 I'll then sketch the new polygon after the translation.

00:52 I'll use this to estimate what the new coordinates of the vertices of the polygon after the translation will be, and then i will actually calculate the coordinates of the vertices of the polygon after the translation.

01:12 So we have to see that for the vertices of our polygon, we'll these range anywhere from 3 up to 7.

01:21 And we're going to be shifting further towards the right.

01:27 So i would say let's range on the x axis from 0 up to 15.

01:42 And for the y -axis, we see that the values lie around 7, but we're going to be shifting down.

01:52 So let's draw the y -axis to range from zero up to 10.

02:10 So we've got these two different axes and we'll plot the points.

02:15 So the point five -eight is around, let's divide this up a little bit more.

02:37 So five -eight is located around here, about halfway between five and ten in the y -axis.

02:43 The next point three -six is located around here.

02:50 The next point, three, next point 76 is located about here.

03:01 So this is our polygon originally.

03:07 Now we're told that the translation tells us to shift six units down and 10 units to left.

03:15 Notice that this translation is rigid in the sense that after translating, our polygon, which is a triangle, will remain a triangle.

03:29 And in fact, the distances between each point of the triangle will remain the same.

03:40 So we can apply the translation to the vertices in order to obtain our new triangle.

03:48 So first, the bottom left vertex, i will shift six units down and ten units to the left.

04:02 Oh, you know what? 10 units to the left.

04:06 My mistake...

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