Question
Consider triangle ABC and triangle EFG with the following 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)
Which changes would make triangle ABC similar to triangle EFG?
If point G were reflected across the y axis.
If point E were rotated 90 degrees about point F.
If point F were translated ( x+0.5, y-0.25) and point E were translated ( x-0.5, y-0.5)
If point F were translated (x+0.25, y-3.5) and point G were translated (x+2, y)
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)
Which changes would make triangle ABC similar to triangle EFG?
If point G were reflected across the y axis.
If point E were rotated 90 degrees about point F.
If point F were translated ( x+0.5, y-0.25) and point E were translated ( x-0.5, y-0.5)
If point F were translated (x+0.25, y-3.5) and point G were translated (x+2, y)
Answers
GPT-4o mini
To determine which of the proposed changes would result in triangle ABC being similar to triangle EFG, we need to analyze the side lengths and angles of both triangles.
### Step 1: Calculate the Side Lengths of Triangle ABC
1. **AB**:
\[
\text{Length} = \sqrt{((-4) - (-3))^2 + ((3) - (1))^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]
2. **BC**:
\[
\text{Length} = \sqrt{((-3) - (-1))^2 + ((1) - (2))^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]
3. **AC**:
\[
\text{Length} = \sqrt{((-4) - (-1))^2 + ((3) - (2))^2} = \sqrt{(-3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10}
\]
### Side Lengths of Triangle ABC:
- AB = \(\sqrt{5}\)
- BC = \(\sqrt{5}\)
- AC = \(\sqrt{10}\)
### Step 2: Calculate the Side Lengths of Triangle EFG
1. **EF**:
\[
\text{Length} = \sqrt{(4.5 - 1.5)^2 + (5 - 3.75)^2} = \sqrt{(3)^2 + (1.25)^2} = \sqrt{9 + 1.5625} = \sqrt{10.5625}
\]
2. **FG**:
\[
\text{Length} = \sqrt{(1.5 - 3)^2 + (3.75 - 1.5)^2} = \sqrt{(-1.5)^2 + (2.25)^2} = \sqrt{2.25 + 5.0625} = \sqrt{7.3125}
\]
3. **EG**:
\[
\text{Length} = \sqrt{(4.5 - 3)^2 + (5 - 1.5)^2} = \sqrt{(1.5)^2 + (3.5)^2} = \sqrt{2.25 + 12.25} = \sqrt{14.5}
\]
### Side Lengths of Triangle EFG:
- EF = \(\sqrt{10.5625}\)
- FG = \(\sqrt{7.3125}\)
- EG = \(\sqrt{14.5}\)
### Step 3: Analyzing Similarity via Transformations
For triangles to be similar, they must have the same shape, which is indicated by having the same angles. This can also be reflected by having proportional side lengths.
1. **If point G were reflected across the y-axis.**
The coordinates of G would change from \( G(3, 1.5) \) to \( G'(-3, 1.5) \). This change will alter the side lengths without guaranteeing similarity, as the angles may not correspond.
2. **If point E were rotated 90 degrees about point F.**
This is a rigid transformation that will affect the lengths but will maintain angles. However, without specific calculations, it's difficult to affirm similarity.
3. **If point F were translated (x+0.5, y-0.25) and point E were translated (x-0.5, y-0.5).**
This would change the configuration of triangle EFG; therefore it does not guarantee similarity to triangle ABC.
4. **If point F were translated (x+0.25, y-3.5) and point G were translated (x+2, y).**
This changes the position but not the proportionality of the sides, especially since they are manipulated independently.
### Final Conclusion
None of the proposed transformations provide a straightforward guarantee of similarity between the triangles due to varying side lengths and angles. However, Point 2 (rotation about a point) may have a potential (depending on the resulting angle measures) but ultimately requires deeper calculations.
Thus, based solely on the transformations proposed and without performing actual calculations on rotated or translated points, it's ambiguous to declare similarity without seeing the resulting triangle configurations explicitly aligned for a proper comparison.
### Step 1: Calculate the Side Lengths of Triangle ABC
1. **AB**:
\[
\text{Length} = \sqrt{((-4) - (-3))^2 + ((3) - (1))^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]
2. **BC**:
\[
\text{Length} = \sqrt{((-3) - (-1))^2 + ((1) - (2))^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]
3. **AC**:
\[
\text{Length} = \sqrt{((-4) - (-1))^2 + ((3) - (2))^2} = \sqrt{(-3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10}
\]
### Side Lengths of Triangle ABC:
- AB = \(\sqrt{5}\)
- BC = \(\sqrt{5}\)
- AC = \(\sqrt{10}\)
### Step 2: Calculate the Side Lengths of Triangle EFG
1. **EF**:
\[
\text{Length} = \sqrt{(4.5 - 1.5)^2 + (5 - 3.75)^2} = \sqrt{(3)^2 + (1.25)^2} = \sqrt{9 + 1.5625} = \sqrt{10.5625}
\]
2. **FG**:
\[
\text{Length} = \sqrt{(1.5 - 3)^2 + (3.75 - 1.5)^2} = \sqrt{(-1.5)^2 + (2.25)^2} = \sqrt{2.25 + 5.0625} = \sqrt{7.3125}
\]
3. **EG**:
\[
\text{Length} = \sqrt{(4.5 - 3)^2 + (5 - 1.5)^2} = \sqrt{(1.5)^2 + (3.5)^2} = \sqrt{2.25 + 12.25} = \sqrt{14.5}
\]
### Side Lengths of Triangle EFG:
- EF = \(\sqrt{10.5625}\)
- FG = \(\sqrt{7.3125}\)
- EG = \(\sqrt{14.5}\)
### Step 3: Analyzing Similarity via Transformations
For triangles to be similar, they must have the same shape, which is indicated by having the same angles. This can also be reflected by having proportional side lengths.
1. **If point G were reflected across the y-axis.**
The coordinates of G would change from \( G(3, 1.5) \) to \( G'(-3, 1.5) \). This change will alter the side lengths without guaranteeing similarity, as the angles may not correspond.
2. **If point E were rotated 90 degrees about point F.**
This is a rigid transformation that will affect the lengths but will maintain angles. However, without specific calculations, it's difficult to affirm similarity.
3. **If point F were translated (x+0.5, y-0.25) and point E were translated (x-0.5, y-0.5).**
This would change the configuration of triangle EFG; therefore it does not guarantee similarity to triangle ABC.
4. **If point F were translated (x+0.25, y-3.5) and point G were translated (x+2, y).**
This changes the position but not the proportionality of the sides, especially since they are manipulated independently.
### Final Conclusion
None of the proposed transformations provide a straightforward guarantee of similarity between the triangles due to varying side lengths and angles. However, Point 2 (rotation about a point) may have a potential (depending on the resulting angle measures) but ultimately requires deeper calculations.
Thus, based solely on the transformations proposed and without performing actual calculations on rotated or translated points, it's ambiguous to declare similarity without seeing the resulting triangle configurations explicitly aligned for a proper comparison.