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Komok [63]
2 years ago
6

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 tra

nsformations (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.
Mathematics
2 answers:
Snezhnost [94]2 years ago
7 0

Answer:

The angle measures are the same.

Step-by-step explanation:

olga55 [171]2 years ago
6 0

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

<h3>How can the new coordinates be found?</h3>

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

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

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;

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

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;

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

  • A(2, 3) → A'(-2, -2)

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

  • B(5, 4) → B'(4, 0)

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

  • 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 <em>x </em>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:

brainly.com/question/12561082

#SPJ1

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2 Here are two equations:
MatroZZZ [7]

a. (3,4) is only the solution to Equation 1.

(4, 2.5) is the solution to both equations

(5,5) is the solution to Equation 2

(3,2) is not the solution to any equation.

b. No, it is not possible to have more than one (x,y) pair that is solution to both equations

Step-by-step explanation:

a. Decide whether each neither of the equations,

i (3,4)

ii. (4,2.5)

ill. (5,5)

iv. (3,2)

To decide whether each point is solution to equations or not we will put the point in the equations

Equations are:

Equation 1: 6x + 4y = 34

Equation 2: 5x – 2y = 15

<u>i (3,4) </u>

Putting in Equation 1:

6(3) + 4(4) = 34\\18+16=34\\34=34\\

Putting in Equation 2:

5(3) - 2(4) = 15\\15-8 = 15\\7\neq 15

<u>ii. (4,2.5)</u>

Putting in Equation 1:

6(4) + 4(2.5) = 34\\24+10=34\\34=34\\

Putting in Equation 2:

5(4) - 2(2.5) = 15\\20-5 = 15\\15=15

<u>ill. (5,5)</u>

6(5) + 4(5) = 34\\30+20=34\\50\neq 34

Putting in Equation 2:

5(5) - 2(5) = 15\\25-10 = 15\\15=15

<u>iv. (3,2)</u>

6(3) + 4(2) = 34\\18+8=34\\26\neq 34

Putting in Equation 2:

5(3) - 2(2) = 15\\15-4 = 15\\11\neq 15

Hence,

(3,4) is only the solution to Equation 1.

(4, 2.5) is the solution to both equations

(5,5) is the solution to Equation 2

(3,2) is not the solution to any equation.

b. Is it possible to have more than one (x, y) pair that is a solution to both

equations?

The simultaneous linear equations' solution is the point on which the lines intersect. Two lines can intersect only on one point. So a linear system cannot have more than one point as a solution

So,

a. (3,4) is only the solution to Equation 1.

(4, 2.5) is the solution to both equations

(5,5) is the solution to Equation 2

(3,2) is not the solution to any equation.

b. No, it is not possible to have more than one (x,y) pair that is solution to both equations

Keywords: Linear equations, Ordered pairs

Learn more about linear equations at:

  • brainly.com/question/10534381
  • brainly.com/question/10538663

#LearnwithBrainly

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