Question
Consider ΔABC , with vertices A(0,3) , B(0,1) , and C(−4,2) , and ΔDEF , with vertices D(0,6) , E(0,2) , and F(8,4) . Which transformations could be used to show that the triangles are similar?
Answers
Answered by
GPT-4o mini
To determine if triangles ΔABC and ΔDEF are similar, we can analyze the transformations that could be applied to one triangle to get the other.
First, let's find the lengths of the sides of each triangle to compare their proportions:
**For ΔABC:**
- Side AB: \( A(0,3) \) and \( B(0,1) \)
- Length \( AB = |3 - 1| = 2 \)
- Side BC: \( B(0,1) \) and \( C(-4,2) \)
- Length \( BC = \sqrt{((-4 - 0)^2 + (2 - 1)^2)} = \sqrt{16 + 1} = \sqrt{17} \)
- Side AC: \( A(0,3) \) and \( C(-4,2) \)
- Length \( AC = \sqrt{((-4 - 0)^2 + (2 - 3)^2)} = \sqrt{16 + 1} = \sqrt{17} \)
Thus, the side lengths of ΔABC are \( AB = 2 \), \( BC = \sqrt{17} \), and \( AC = \sqrt{17} \).
**For ΔDEF:**
- Side DE: \( D(0,6) \) and \( E(0,2) \)
- Length \( DE = |6 - 2| = 4 \)
- Side EF: \( E(0,2) \) and \( F(8,4) \)
- Length \( EF = \sqrt{((8 - 0)^2 + (4 - 2)^2)} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17} \)
- Side DF: \( D(0,6) \) and \( F(8,4) \)
- Length \( DF = \sqrt{((8 - 0)^2 + (4 - 6)^2)} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17} \)
Thus, the side lengths of ΔDEF are \( DE = 4 \), \( EF = 2\sqrt{17} \), and \( DF = 2\sqrt{17} \).
**Next, let's analyze the corresponding sides:**
- For the side lengths, we can compare the ratios:
- The ratio of side AB to side DE is \( \frac{2}{4} = \frac{1}{2} \).
- The ratio of side BC to side EF and AC to side DF is \( \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2} \).
Since all corresponding sides are proportional at the same ratio of \( \frac{1}{2} \), the triangles are similar.
**The transformations that can be used to show that triangles are similar are:**
1. **Scaling (Dilation):** A dilation with a scale factor of \( \frac{1}{2} \) from the center of the triangle can transform ΔDEF into ΔABC.
2. **Rotation/Reflection:** Depending on the orientation of the triangles relative to each other, a rotation or reflection may also be necessary to align the triangles correctly before applying the dilation.
**Conclusion:** Triangles ΔABC and ΔDEF are similar by a sequence of transformations that may include dilation, along with possible rotation or reflection.
First, let's find the lengths of the sides of each triangle to compare their proportions:
**For ΔABC:**
- Side AB: \( A(0,3) \) and \( B(0,1) \)
- Length \( AB = |3 - 1| = 2 \)
- Side BC: \( B(0,1) \) and \( C(-4,2) \)
- Length \( BC = \sqrt{((-4 - 0)^2 + (2 - 1)^2)} = \sqrt{16 + 1} = \sqrt{17} \)
- Side AC: \( A(0,3) \) and \( C(-4,2) \)
- Length \( AC = \sqrt{((-4 - 0)^2 + (2 - 3)^2)} = \sqrt{16 + 1} = \sqrt{17} \)
Thus, the side lengths of ΔABC are \( AB = 2 \), \( BC = \sqrt{17} \), and \( AC = \sqrt{17} \).
**For ΔDEF:**
- Side DE: \( D(0,6) \) and \( E(0,2) \)
- Length \( DE = |6 - 2| = 4 \)
- Side EF: \( E(0,2) \) and \( F(8,4) \)
- Length \( EF = \sqrt{((8 - 0)^2 + (4 - 2)^2)} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17} \)
- Side DF: \( D(0,6) \) and \( F(8,4) \)
- Length \( DF = \sqrt{((8 - 0)^2 + (4 - 6)^2)} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17} \)
Thus, the side lengths of ΔDEF are \( DE = 4 \), \( EF = 2\sqrt{17} \), and \( DF = 2\sqrt{17} \).
**Next, let's analyze the corresponding sides:**
- For the side lengths, we can compare the ratios:
- The ratio of side AB to side DE is \( \frac{2}{4} = \frac{1}{2} \).
- The ratio of side BC to side EF and AC to side DF is \( \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2} \).
Since all corresponding sides are proportional at the same ratio of \( \frac{1}{2} \), the triangles are similar.
**The transformations that can be used to show that triangles are similar are:**
1. **Scaling (Dilation):** A dilation with a scale factor of \( \frac{1}{2} \) from the center of the triangle can transform ΔDEF into ΔABC.
2. **Rotation/Reflection:** Depending on the orientation of the triangles relative to each other, a rotation or reflection may also be necessary to align the triangles correctly before applying the dilation.
**Conclusion:** Triangles ΔABC and ΔDEF are similar by a sequence of transformations that may include dilation, along with possible rotation or reflection.