Question Video: Translating a Shape Using Its Coordinates Mathematics

Which of the following represents the image of △𝐴𝐡𝐢, where 𝐴(1, 2), 𝐡(1, 5), and 𝐢(3, 5), after a transformation (π‘₯, 𝑦) β†’ (π‘₯ + 3, 𝑦 βˆ’ 2)? [A] 𝐴′(4, 2), 𝐡′(4, 5), and 𝐢′(6, 5) [B] 𝐴′(1, 0), 𝐡′(1, 3), and 𝐢′(3, 3) [C] 𝐴′(4, 0), 𝐡′(4, 3), and 𝐢′(6, 3) [D] 𝐴′(βˆ’1, 5), 𝐡′(βˆ’1, 8), and 𝐢′(1, 8)


Video Transcript

Which of the following represents the image of triangle 𝐴𝐡𝐢, where 𝐴 has coordinates one, two; 𝐡 has coordinates one, five; and 𝐢 has coordinates three, five, after a transformation that maps π‘₯, 𝑦 onto π‘₯ plus three, 𝑦 minus two? Option (A) 𝐴 prime four, two; 𝐡 prime four, five; and 𝐢 prime six, five. Option (B) 𝐴 prime one, zero; 𝐡 prime one, three; and 𝐢 prime three, three. Option (C) 𝐴 prime four, zero; 𝐡 prime four, three; and 𝐢 prime six, three. Or option (D) 𝐴 prime negative one, five; 𝐡 prime negative one, eight; and 𝐢 prime one, eight.

We’re given the rule that describes this transformation. Every point π‘₯, 𝑦 is mapped to the point π‘₯ plus three, 𝑦 minus two. We note that this transformation is a translation. A translation of an object is when the object is moved by sliding it by a set number of units vertically and horizontally. According to the given rule, the triangle will be translated three units to the right and two units down.

To find out which of the options given represents the image of triangle 𝐴𝐡𝐢, we substitute the coordinates of each of the vertices of triangle 𝐴𝐡𝐢 into the translation rule to give us the vertices of the image. The point 𝐴 with coordinates one, two is mapped to the point 𝐴 prime. The coordinates of 𝐴 prime are found by adding three to the π‘₯-coordinate one and subtracting two from the 𝑦-coordinate two. So, 𝐴 prime has coordinates four, zero. According to the same rule, point 𝐡 with coordinates one, five is mapped to the point 𝐡 prime with coordinates four, three. And finally, the point 𝐢 with coordinates three, five is mapped to point 𝐢 prime with coordinates six, three, as shown.

Looking carefully at the four options given, we can see that this set of coordinates is option (C). In conclusion, we found the coordinates of the image of triangle 𝐴𝐡𝐢 by applying the translation rule to each vertex.

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