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A triangle on a coordinate plane has three vertices A(2 , 3), B(5, 4), and C(3, 6). Use this description to do the following transformations (if needed, draw this triangle on a sheet of paper):

a. Dilation 1: What would be the new coordinates if this triangle were dilated to a scale factor of 2 with the center of the dilation at the origin? How did you determine these points?
b. Dilation 2: What would be the new coordinates if this triangle were dilated to a scale factor of 2 with the center of the dilation at the point (6, 8)? How did you determine these points?
c. What series of transformations would carry dilation 1 onto dilation 2? Compare Dilation 1 to Dilation 2. Explain what conclusions you can draw about the scale factor, difference in area, and center of dilation.
d. What is the proportion of the side lengths from Dilation 1 to Dilation 2? What is the proportion of their angle measures? Explain your answer.

Respuesta :

The dilation by a scale factor of 2 of the points A(2, 3), B(5, 4), C(3, 6) gives;

a. A'(4, 6), B'(10, 8), C'(6, 12)

b. A'(-2, -2), B'(4, 0), C'(0, 4)

c. The transformation that would carry dilation 1 onto dilation 2 is T(-6, -8)

  • The area of dilation 1 and 2 are the same
  • The center of dilation does not change the area

d. The proportion of the side length of Dilation 1 and Dilation 2 is 1:1

  • The angle measures are the same

How can the new coordinates be found?

The general formula for finding the coordinates of the image of a point following a dilation is presented as follows;

[tex]D _{(a , \: b)k}(x, \: y) = (a + k \times (x - a) , \: b+ k \times (y - b))[/tex]

Where;

(a, b) = The center of dilation

k = The scale factor of dilation

(x, y) = The coordinate of the pre-image

The given points are;

A(2, 3), B(5, 4), C(3, 6)

a. The scale factor of dilation = 2

The center of dilation = The origin (0, 0)

Therefore;

[tex]D _{(0 , \: 0)2}(2, \: 3) = (0 + 2 \times (2 - 0) , \: 0+ 2 \times (3 - 0)) = (4, \:6)[/tex]

Therefore dilation about the origin, with a scale factor of 2 gives;

  • A(2, 3) → A'(4, 6)

Similarly

  • B(5, 4) → B'(10, 8)

  • C(3, 6) → C'(6, 12)

b. With the center of dilation at (6, 8), we have;

[tex]D _{(6 , \: 8)2}(2, \: 3) = (6 + 2 \times (2 - 6) , \: 8+ 2 \times (3 - 8)) = (-2, \:-2)[/tex]

  • A(2, 3) → A'(-2, -2)

[tex]D _{(6 , \: 8)2}(5, \: 4) = (6 + 2 \times (5 - 6) , \: 8+ 2 \times (4 - 8)) = (4, \:0)[/tex]

  • B(5, 4) → B'(4, 0)

[tex]D _{(6 , \: 8)2}(3, \: 6) = \mathbf{(6 + 2 \times (3 - 6) , \: 8+ 2 \times (6 - 8))} = (0, \:4)[/tex]

  • C(3, 6) → C'(0, 4)

c. The difference between the coordinates of the points on dilation 1 and 2 is a shift left 6 places and a shift downwards 8 places

Using notation, we have;

  • Dilation 1 T(-6, -8) → Dilation 2

The area of the images of dilation 1 and 2 are equal given that the scale factor is the same.

  • The location of the center of dilation does not change the area of the image

d. From the above calculation, given that the difference between pre-image point and the center is multiplied by the scale factor followed by the addition of the x and y-values, the lengths of the sides of dilation 1 and 2 are the same, such that we have;

  • The proportion of the side lengths is 1

Given that the side lengths are the same, by AAA congruency postulate, we have;

  • The angle measures are the same.

Learn more about dilation transformation here:

https://brainly.com/question/12561082

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CookieNeedsAnswers

Answer:

The angle measures are the same.

Step-by-step explanation:

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