To determine why triangles RST and DEF are similar, we need to examine the relationships between their corresponding angles and sides.
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Ratio of Sides: We can calculate the lengths of the sides of each triangle using the distance formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For Triangle RST:
- Length of RS: \[ RS = \sqrt{(0 - (-2))^2 + (2 - (-1))^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \]
- Length of ST: \[ ST = \sqrt{(4 - 0)^2 + (-1 - 2)^2} = \sqrt{(4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
- Length of RT: \[ RT = \sqrt{(4 - (-2))^2 + (-1 - (-1))^2} = \sqrt{(6)^2 + (0)^2} = \sqrt{36} = 6 \]
For Triangle DEF:
- Length of DE: \[ DE = \sqrt{(0 - (-4))^2 + (4 - (-2))^2} = \sqrt{(4)^2 + (6)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} \]
- Length of EF: \[ EF = \sqrt{(8 - 0)^2 + (-2 - 4)^2} = \sqrt{(8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]
- Length of DF: \[ DF = \sqrt{(8 - (-4))^2 + (-2 - (-2))^2} = \sqrt{(12)^2 + (0)^2} = \sqrt{144} = 12 \]
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Comparing Ratios: Now we can check if the sides are proportional:
- Ratio of RS to DE: \[ \frac{RS}{DE} = \frac{\sqrt{13}}{2\sqrt{13}} = \frac{1}{2} \]
- Ratio of ST to EF: \[ \frac{ST}{EF} = \frac{5}{10} = \frac{1}{2} \]
- Ratio of RT to DF: \[ \frac{RT}{DF} = \frac{6}{12} = \frac{1}{2} \]
Since all corresponding sides are in the same ratio of 1:2, the triangles are similar by the Side-Side-Side (SSS) similarity criterion.
- Transformation: The transformation that relates triangle RST to triangle DEF is a dilation. Specifically, triangle DEF is a result of dilating triangle RST by a scale factor of 2, about the origin (or any point).
Thus, we can conclude that: Triangle RST is similar to triangle DEF because the lengths of their corresponding sides are proportional (1:2), and the transformation from RST to DEF is a dilation with a scale factor of 2.
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