Graph FGH with vertices F(□2,2), G(□2, □4), and H(□4, □4) and its image after the similarity tra

step 1 So let's look at composite transformation. So let's say we wanted to translate and dilate a tria

Graph FGH with vertices F(□2,2), G(□2, □4), and H(□4, □4)...

Graph $\triangle \mathrm{FGH}$ with vertices $\mathrm{F}(\square 2,2),$ $\mathrm{G}(\square 2, \square 4),$ and $\mathrm{H}(\square 4, \square 4)$ and its image after the similarity transformation. ( See Example 1.) Translation: $(\mathrm{x}, \mathrm{y}) \rightarrow(\mathrm{x}+3, \mathrm{y}+1)$ Dilation: $(\mathrm{x}, \mathrm{y}) \rightarrow(2 \mathrm{x}, 2 \mathrm{y})$

Graph $\triangle \mathrm{FGH}$ with vertices $\mathrm{F}(\square 2,2),$ $\mathrm{G}(\square 2, \square 4),$ and $\mathrm{H}(\square 4, \square 4)$ and its image after the similarity transformation. ( See Example 1.)
Translation: $(\mathrm{x}, \mathrm{y}) \rightarrow(\mathrm{x}+3, \mathrm{y}+1)$
Dilation: $(\mathrm{x}, \mathrm{y}) \rightarrow(2 \mathrm{x}, 2 \mathrm{y})$

Key Concepts

Dilation

Dilation is a transformation that enlarges or reduces a figure by a scale factor relative to a fixed point, often the origin. It changes the size of the figure while preserving the shape and the proportional relationships between its parts. Each coordinate of the original figure is multiplied by the scale factor, leading to an enlarged or reduced image.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometric figures using a coordinate system. It enables the analysis and transformation of shapes through algebraic methods, such as translations and dilations, by applying operations directly to their coordinate points.

Similarity Transformation

A similarity transformation is a combination of operations that produces an image that is similar to the original figure. This means the image has the same shape but can differ in size and position. Similarity transformations typically involve dilations, translations, rotations, and reflections, which preserve the overall proportions and angles of the original figure.

Translation

Translation is a type of transformation that moves every point of a figure the same distance in the same direction. In the coordinate system, this is achieved by adding a constant value to the x-coordinate and a constant value to the y-coordinate, resulting in a new position without changing the shape or orientation of the figure.

Transcript

00:03 So let's look at composite transformation.

00:06 So let's say we wanted to translate and dilate a triangle.

00:14 All right.

00:15 I'm going to do the translation in blue.

00:16 So let's say we want to translate the triangle here, right? x plus 3 and y plus 1.

00:33 And then the second transformation is going to be the dilation.

00:36 And we're going to multiply by 2.

00:38 Right so when we write the rule out we're going to get 2x and 2 y all right and then we need to know what our vertices are for this triangle that we are going to transform and so we have f at negative 2 2 and then g and then g and at negative 2, negative 4, and then h at negative 4, negative 4.

01:24 All right.

01:25 So like i said, translation isn't blue, dilation is in red.

01:28 And so now when we translate, we're just going to essentially do the math.

01:35 And so f prime here, right? we're going to add three to the x values and then add one to the y value.

01:44 And so negative 2 plus 3 is going to give me a positive 1 and then 2 plus 1 is 3 g prime again negative 2 plus 3 is still 1 negative 4 plus 1 is a negative 3 and then for h we get h prime is negative 4 plus 3 which is a negative 1 and then negative 4 plus 1 which is negative three.

02:23 All right.

02:24 So that is the translation.

02:27 And so now we have to dilate the triangle.

02:31 And again, the dilation tells us to multiply everything by two.

02:36 And so here, right, when we multiply, we're going to multiply the new figure after the translation.

02:46 And so f prime is going to become f prime.

02:50 Prime all right and then we just multiply by two so one times two is two three times two is six and then g prime prime we're going to get one times two is two and then negative three times two is negative six and then h double prime right negative one times two is a negative two and then negative three times 2 is negative 6 and that would be right our image right so the triangle at the beginning right i wrote that in black that was our pre -image and then our image or our final answer is in red and so right normally they would want us to graph questions like this right graph the original triangle and its image.

03:54 And so we can go ahead and we could graph that here.

04:00 Right...

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