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Consider triangle ABC, with vertices A(0, 3), B(0, 1), and C(-4, 2), and triangle DEF, with vertices D(0, 6), E(0, 2), and F(8, 4). Which transformations could be used to show that the triangles are similar?

1 answer

To determine if triangles ABC and DEF are similar, we will first check the corresponding angles and the ratios of their sides.

  1. List the coordinates:

    • Triangle ABC has vertices:

      • \( A(0, 3) \)
      • \( B(0, 1) \)
      • \( C(-4, 2) \)

    • Triangle DEF has vertices:

      • \( D(0, 6) \)
      • \( E(0, 2) \)
      • \( F(8, 4) \)

  2. Calculate the lengths of the sides of triangle ABC:

    • AB: \[ AB = \sqrt{(0 - 0)^2 + (3 - 1)^2} = \sqrt{0 + 4} = 2 \]
    • AC: \[ AC = \sqrt{(0 - (-4))^2 + (3 - 2)^2} = \sqrt{(4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17} \]
    • BC: \[ BC = \sqrt{(0 - (-4))^2 + (1 - 2)^2} = \sqrt{(4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \]

    Thus, the sides of triangle ABC are \( AB = 2 \), \( AC = \sqrt{17} \), and \( BC = \sqrt{17} \).

  3. Calculate the lengths of the sides of triangle DEF:

    • DE: \[ DE = \sqrt{(0 - 0)^2 + (6 - 2)^2} = \sqrt{0 + 16} = 4 \]
    • DF: \[ DF = \sqrt{(0 - 8)^2 + (6 - 4)^2} = \sqrt{(8)^2 + (2)^2} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17} \]
    • EF: \[ EF = \sqrt{(8 - 0)^2 + (4 - 2)^2} = \sqrt{(8)^2 + (2)^2} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17} \]

    Thus, the sides of triangle DEF are \( DE = 4 \), \( DF = 2\sqrt{17} \), and \( EF = 2\sqrt{17} \).

  4. Check the ratio of corresponding sides:

    • For side \( AB \) and \( DE \): \[ \frac{AB}{DE} = \frac{2}{4} = \frac{1}{2} \]
    • For side \( AC \) and \( DF \): \[ \frac{AC}{DF} = \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2} \]
    • For side \( BC \) and \( EF \): \[ \frac{BC}{EF} = \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2} \]

  5. Conclusion about similarity: Since the ratios of corresponding sides of triangles ABC and DEF are all equal to \( \frac{1}{2} \), triangle ABC is similar to triangle DEF by the Side-Side-Side (SSS) similarity criterion.

  6. Transformation: The transformation from triangle ABC to triangle DEF involves a scaling (dilation) with a scale factor of \( 2 \), along with possible translations to align the triangles, since both triangles can also be repositioned in the coordinate plane.

Thus, a scaling transformation followed by possible translations can show that triangles ABC and DEF are similar.