Question Video: Finding the Measure of the Angle of Rotation Mathematics • 8th Grade

△𝐴𝐵′𝐶′ is the image of △𝐴𝐵𝐶 by a counterclockwise rotation of 𝑥° about 𝐴. Find 𝑥.

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Video Transcript

Triangle 𝐴 𝐵 prime 𝐶 prime is the image of triangle 𝐴𝐵𝐶 by a counterclockwise rotation of 𝑥 degrees about 𝐴. Find 𝑥.

So, in this question, we’re given that there’s a rotation. Triangle 𝐴𝐵𝐶 is rotated, and we’re told that the direction is counterclockwise, to give us triangle 𝐴 𝐵 prime 𝐶 prime. The angle of rotation is 𝑥 degrees, and we need to work out what 𝑥 is.

The center of rotation of this rotation is at vertex 𝐴, which explains why there’s no new image of 𝐴 of 𝐴 prime. In order to work out the angle of rotation between an object and its image, we can calculate the angle between any vertex and the image of that vertex. Let’s begin with vertex 𝐵. We know that it’s rotated counterclockwise to the image 𝐵 prime. If we joined each vertex, 𝐵 and 𝐵 prime, with a line to the center of rotation, then to find the angle of rotation, we just need to find the angle between these two lines.

So, 𝑥, the angle of rotation, is equal to 37 degrees plus 69 degrees, which means that 𝑥 is equal to 106. We can check our answer by finding the angle between 𝐶 and 𝐶 prime. To find the angle here, once again we’ll be adding 69 degrees and 37 degrees, which gives us an answer of 106.

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