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# How do Similar Triangles in the XY-Coordinate Plane Calculate Image Changes?

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▲ good image

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Question: In the xy-coordinate plane, Δ ABC has vertices at A(1, -2), B(1, 0.5), and C(2, 1) and Δ DEF has vertices at D(4, -3), E(4, 2), and F(6, 3). The triangles are similar because Δ DEF is the image of Δ ABC under a dilation.

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Since both triangles are given as being similar, which means they have the same angles and their sides are proportional, let's first verify the angles. To do this, we can calculate the slopes of the lines that form these triangles sides: Slope of AB = (0.5 - (-2)) / (1 - 1) = 2.5 / 0 = Undefined (This means that AB is a vertical line) Slope of AC = (1 - (-2)) / (2 - 1) = 3 / 1 = 3 Slope of BC = (0.5 - 1) / (1 - 2) = (-0.5) / (-1) = 0.5 Now, let's find out the slopes of ΔDEF: Slope of DE = (2 - (-3)) / (4 - 4) = 5 / 0 = Undefined (This means that DE is a vertical line) Slope of DF = (3 - (-3)) / (6 - 4) = 6 / 2 = 3 Slope of EF = (2 - 3) / (4 - 6) = (-1) / (-2) = 0.5 The slopes of corresponding sides are the same, so the triangles have the same angles. Now let's verify if they are proportional: Length of AB = |0.5 - (-2)| = 2.5 Length of AC = √((2 - 1)² + (1 - (-2))²) = √(1 + 9) = √10 Length of BC = √((1 - 2)² + (0.5 - 1)²) = √(1 + 0.25) = √1.25 Length of DE = |2 - (-3)| = 5 Length of DF = √((6 - 4)² + (3 - (-3))²) = √(4 + 36) = √40 Length of EF = √((4 - 6)² + (2 - 3)²) = √(4 + 1) = √5 Now let's find the ratio of the sides: Length of DE / Length of AB = 5 / 2.5 = 2 Length of DF / Length of AC = √40 / √10 = 2 (Since √40 = 2√10) Length of EF / Length of BC = √5 / √1.25 = 2 (Since √5 = √(4 × 1.25)) Since the ratios of corresponding side lengths are equal (2:1 in this case), ΔDEF is indeed the image of ΔABC under a dilation, and the two triangles are similar.

June 21, 2023, 10:40 a.m.

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