Use the image to answer the question.

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

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
Skip to navigation

Answer 1

To determine which transformations would make triangle \( ABC \) similar to triangle \( EFG \), we need to check the scale factors and angles of the triangles. Triangles are similar if their corresponding angles are equal and their corresponding side lengths are proportional.

1. Checking the vertices of \( \triangle ABC \) and \( \triangle EFG \):

   • For \( \triangle ABC \):
     • A(-4, 3), B(-3, 1), C(-1, 2)
   • For \( \triangle EFG \):
     • E(4.5, 5), F(1.5, 3.75), G(3, 1.5)

2. Calculating the lengths of the sides for both triangles:

   • Length of sides \( AB \), \( BC \), and \( AC \) for \( \triangle ABC \):

     • \( AB = \sqrt{((-4) - (-3))^2 + (3 - 1)^2} = \sqrt{1 + 4} = \sqrt{5} \)
     • \( BC = \sqrt{((-3) - (-1))^2 + (1 - 2)^2} = \sqrt{4 + 1} = \sqrt{5} \)
     • \( AC = \sqrt{((-4) - (-1))^2 + (3 - 2)^2} = \sqrt{9 + 1} = \sqrt{10} \)

   • Length of sides \( EF \), \( FG \), and \( EG \) for \( \triangle EFG \):

     • \( EF = \sqrt{(4.5 - 1.5)^2 + (5 - 3.75)^2} = \sqrt{(3)^2 + (1.25)^2} = \sqrt{9 + 1.5625} = \sqrt{10.5625} \)
     • \( FG = \sqrt{(1.5 - 3)^2 + (3.75 - 1.5)^2} = \sqrt{(-1.5)^2 + (2.25)^2} = \sqrt{2.25 + 5.0625} = \sqrt{7.3125} \)
     • \( EG = \sqrt{(4.5 - 3)^2 + (5 - 1.5)^2} = \sqrt{(1.5)^2 + (3.5)^2} = \sqrt{2.25 + 12.25} = \sqrt{14.5} \)

3. Finding a transformation that would make the triangles similar:

   • Similarity can often be achieved through rotation, reflection, or translation. However, we need to ensure that the angles remain the same, and the sides are proportional.

4. Evaluating the options:

   • Rotation of E and Translation of F and E are likely candidates along with reflection of G, but we need to confirm proportionality.
     • Neither transformation should distort the angles; they need to maintain the same shape.
     • We’ll analyze the effects of the proposed transformations.

Looking at the options, it appears that:

If point G is reflected across the y-axis, it could potentially align the points of triangle \( ABC \) with triangle \( EFG \) in a way that aligns their shapes.

Thus, the correct response is: if point G were reflected across the y-axis.