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Question

Triangle A B C and triangle G E F are graphed on a coordinate plane. Both the axes range from negative 5 to 5 in unit increments.

Consider ΔABC and ΔEFG with the following vertices:

ΔABC , A(−4,3) , B(−3,1) , C(−1,2)

ΔEFG , E(4.5,5) , F(1.5,3.75) , G(3,1.5)

Which changes would make ΔABC similar to ΔEFG ?

(1 point)
Responses which answer is correct and why by proving it

if point E were rotated 90 degrees about point F
if point upper E were rotated 90 degrees about point upper D

if point F were translated (x+0.5,y−0.25) and point E were translated (x−0.5,y−0.5)
if point upper F were translated left parenthesis x plus 0.5 comma y minus 0.25 right parenthesis and point upper E were translated left parenthesis x minus 0.5 comma y minus 0.5 right parenthesis

if point G were reflected across the y-axis
if point upper G were reflected across the y -axis

if point F were translated (x+0.25,y−3.5)and point G were translated (x+2,y)
if point upper F were translated left parenthesis x plus 0.25 comma y minus 3.5 right parenthesis and point upper G were translated left parenthesis x plus 2 comma y right parenthesis
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Answer 1

To determine which changes would make triangle ABC similar to triangle EFG, we need to check for transformations like congruence (which maintain angles) and proportionate side lengths (which also maintain the similarity).

First, let's analyze the two triangles based on their vertices:

Vertices:

• Triangle ABC:

  • A(−4,3)
  • B(−3,1)
  • C(−1,2)

• Triangle EFG:

  • E(4.5,5)
  • F(1.5,3.75)
  • G(3,1.5)

Steps to Analyze Similarity:

1. Calculating Side Lengths:

   • For triangle ABC:

     • AB = √[(-3 - (-4))^2 + (1 - 3)^2] = √[1^2 + (-2)^2] = √5
     • BC = √[(-1 - (-3))^2 + (2 - 1)^2] = √[2^2 + 1^2] = √5
     • AC = √[(-1 - (-4))^2 + (2 - 3)^2] = √[3^2 + (-1)^2] = √10

   • For triangle EFG:

     • EF = √[(1.5 - 4.5)^2 + (3.75 - 5)^2] = √[(-3)^2 + (-1.25)^2] = √[9 + 1.5625] = √10.5625
     • FG = √[(3 - 1.5)^2 + (1.5 - 3.75)^2] = √[(1.5)^2 + (-2.25)^2] = √[2.25 + 5.0625] = √7.3125
     • EG = √[(3 - 4.5)^2 + (1.5 - 5)^2] = √[(-1.5)^2 + (-3.5)^2] = √[2.25 + 12.25] = √14.5

2. Ratios of sides:

   • Ratio of sides in triangle ABC (using sides calculated):
     • Ratio of AB:BC:AC is √5:√5:√10
   • Ratio of sides in triangle EFG:
     • Ratio of EF:FG:EG is √10.5625:√7.3125:√14.5

To determine which transformations will maintain the proportions and angles, let's analyze the proposed transformations one by one:

Options:

1. If point E were rotated 90 degrees about point F:

   • A rotation doesn’t maintain side lengths and angles. It changes proportions.

2. If point E were rotated 90 degrees about point D:

   • Not applicable since D isn’t defined.

3. Translating points:

   • Translating F (x+0.5, y-0.25) and E (x−0.5, y−0.5):
   • This translation changes the location but does not change the proportions or angles.

4. Translating F (x+0.5, y-0.25) and E (x−0.5, y−0.5):

   • Same as above, does not maintain similarity.

5. If point G were reflected across the y-axis:

   • This point transformation would not maintain proportions.

6. If point F were translated (x+0.25,y−3.5) and point G were translated (x+2,y):

   • This translation changes the triangle location but not the dimension properties.

Conclusion:

None of the transformations described would result in triangle ABC becoming similar to triangle EFG based on the side lengths and angles.

Therefore, none of the transformations provided will make triangle ABC similar to triangle EFG. Similarity needs to maintain the proportion of sides, which none of the given options achieve.