![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-21.png\")
<\/figure>\n\n\n\nAs a refresher, pre-image refers to the original figure and the image is the resulting figure after the Take a look at the two pairs of images shown above. After rotating the pre-images over a reference point, the resulting images are simply the pre-image being flipped over horizontally. Now, what happens when we flip a coordinate or a polygon on a Cartesian plane?<\/p>\n\n\n\n When given a coordinate point, [katex](x, y)[\/katex], when we rotate it a 180o degree rotation with respect to the origin, the resulting point will have coordinates that are the negative equivalents of the original point\u2019s. It\u2019s equivalent to flipping the point over the x-axis then the y-axis. To visualize this better, imagine rotating [katex]A = (4, 4)[\/katex] in a 180-degree rotation with respect to the origin. The resulting point is [katex]A^{\\prime}= (-4, -4)[\/katex] as shown below.<\/p>\n\n\n\n One way to locate the rotated point\u2019s position is to connect the original point, [katex]A =(4,4)[\/katex], with a line passing through the origin. The distance of A and A\u2019 from the origin are equal, so the rotated point\u2019s position is easy to spot. We\u2019ve also learned that the rotated point\u2019s coordinates will simply be the negative equivalents of the pre-image\u2019s coordinates. <\/p>\n\n\n\n [katex] \\begin{aligned}A \\rightarrow A^{\\prime}: (4, 4) \\rightarrow(-4, -4)\\end{aligned} [\/katex]<\/p>\n\n\n\n Let\u2019s now observe how 180 degree rotation affects figures like the rectangle shown below. The transformation shown below also confirms that the angles formed by [katex]A\\rightarrow A^{\\prime}[\/katex], [katex]B\\rightarrow B^{\\prime}[\/katex], [katex]C\\rightarrow C^{\\prime}[\/katex], and [katex]D\\rightarrow D^{\\prime}[\/katex] are all straight angles (meaning, they measure 180o<\/sup>). As expected, the resulting image is equivalent to flipping the rectangle, ABCD, over the x-axis followed by the y-axis.<\/p>\n\n\n\n Take a look at how the four vertices of ABCD are affected by the 180-degree rotation. We\u2019re expecting the vertices\u2019 coordinates to change to their negative equivalents and follow the transformation: [katex](x,y ) \\rightarrow (-x, -y)[\/katex]. <\/p>\n\n\n\n This confirms that when finding the coordinate of the resulting image, simply multiply the pre-image\u2019s coordinates by -1 each. Before moving on to the next section, test your knowledge first by working on the problem that follows. <\/p>\n\n\n\n The point, A, is rotated 180o<\/sup> degrees to project the resulting point, A\u2019. The same goes for the three other pairs of points. <\/p>\n\n\n\n Which of the pairs of points do not reflect the correct result after the point is rotated at 180 degrees?<\/p>\n\n\n\n Observe the angle formed between the segments connecting the origin with the pre-image and image. <\/p>\n\n\n\n From this, it is safe to conclude that C is not rotated properly in a 180o<\/sup> angle with respect to the origin. But there is another way to double-check this- by observing how each of the points\u2019 coordinates change after being rotated at 180o<\/sup>.<\/p>\n\n\n\n The three pairs of points other than [katex]C\\rightarrow C^{\\prime}[\/katex], exhibit the expected behavior of points being rotated at 180 degrees.\u00a0<\/p>\n\n\n\n Now that we\u2019ve established the fundamentals of a 180-degree rotation, it\u2019s time to break down the steps needed to rotate figures.<\/p>\n\n\n\n There are two ways to transform a point or a figure and rotate it 180 degrees with respect to a point or an origin. Use the reference point and construct two rays equidistant from it or apply the rules for coordinates when the figure is graphed on a xy-plane.<\/p>\n\n\n\n The first method is helpful when we\u2019re working with a figure outside the Cartesian plane. When rotating an object at a 180-degree angle with respect to a reference point, flip the object horizontally then vertically. The reference point must still be equidistant from the pre-image and image\u2019 positions.<\/p>\n\n\n\n Apply this method to rotate the pre-image (pentagon) shown below. Describe the resulting image\u2019s shape. <\/p>\n\n\n\n When rotating the thunder symbol at 180 degrees, construct a line segment connecting the pre-image of the hexagon and the reference point. <\/p>\n\n\n\n The reason we\u2019re doing this is that we want to form a straight angle to guide us in finding the resulting image after the 180-degree rotation. Now, connect this with a segment sharing the same length to mark the position of the resulting image. We expect the image to be identical to the pre-image, but this time, it\u2019s flipped over horizontally.<\/p>\n\n\n\n This shows that the resulting image of the hexagon after we\u2019ve performed 180-degree rotation is equivalent to the pre-image being flipped over horizontally then vertically. The shape and size of the figure remain the same \u2013 this means that this transformation is also an example of isometry (transformations that do not affect shape and size).<\/p>\n\n\n\n Now, what if we\u2019re rotating on a Cartesian plane? This process is simpler if we\u2019re given the vertices of the figure. Apply what we\u2019ve learned about the effects of a 180-degree rotation on the coordinates of the figures. By rotating the key points, we\u2019ll have an idea of how the rotation affects the entire graph. Connect the rotated vertices to construct the resulting image.<\/p>\n\n\n\n Let\u2019s try out this method to rotate a line passing through the points, A = (4, 5) and B = (-6, -5), by 180o<\/sup> degrees. <\/p>\n\n\n\n We only need two points to graph a line, so to find the resulting line after the 180-degree rotation, we only need to focus on the two given points. As we have mentioned, multiply the pre-image\u2019s coordinates by -1 to find the resulting coordinates of the image after the 180-degree rotation. <\/p>\n\n\n\n [katex]\\begin{aligned}A = (4, 5) &\\rightarrow A^{\\prime}= (-4, -5)\\\\ B = (-6, -5) &\\rightarrow A^{\\prime}= (6, 5)\\end{aligned}[\/katex]<\/p>\n\n\n\n Now that we have the coordinates for the rotated points, simply connect the two to form the line segment. Check out the two lines shown above \u2013 highlighting the effect of a 180-degree rotation on AB. <\/em><\/p>\n\n\n\n Use a similar process to rotate the triangle, [katex]\\Delta ABC[\/katex], as shown below. Extend the xy-plane and construct the resulting image after a 180-degree transformation.<\/p>\n\n\n\n When rotating a point or a figure on the xy-plane, take note of the important vertices. For the case of [katex]\\Delta ABC[\/katex], we have the following coordinates of its vertices:<\/p>\n\n\n\n [katex]\\begin{aligned}A &= (1, 0)\\\\B&= (5, 4)\\\\C &= (8, 0)\\end{aligned}[\/katex]<\/p>\n\n\n\n When applying a 180 degree rotation on a point, simply multiply its coordinates by -1. Apply the same process for all three vertices of the triangle. These rotated points will be the vertices of the resulting image. <\/p>\n\n\n\n Now that we have the vertices for the rotated triangle, connect all three to form the rotated triangle, [katex]\\Delta A^{\\prime}B^{\\prime}C^{\\prime}[\/katex].<\/p>\n\n\n\n The image of the triangle will now have vertices at the following points: (-1, 0), (-5, -4), and (-8, 0). Looking at the two triangles, we can also confirm that the resulting triangle is simply equivalent to flipping the pre-image over the x-axis then over the y-axis.<\/p>\n","protected":false},"excerpt":{"rendered":" The 180-degree rotation (both clockwise and counterclockwise) is one of the simplest and most used transformations in geometry. In this article, we want you to understand what makes this transformation unique, its fundamentals, and understand the two important methods we can use to rotate a figure 180 degrees (in either direction).<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-434","post","type-post","status-publish","format-standard","hentry","category-geometry"],"_links":{"self":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/434","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/comments?post=434"}],"version-history":[{"count":9,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/434\/revisions"}],"predecessor-version":[{"id":459,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/434\/revisions\/459"}],"wp:attachment":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/media?parent=434"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/categories?post=434"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/tags?post=434"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Original Point (Pre-image)<\/strong><\/td> Rotated Point (Image)<\/strong><\/td><\/tr> [katex]\\begin{aligned}(x, y)\\end{aligned}[\/katex]<\/td> [katex]\\begin{aligned}(-x, -y)\\end{aligned}[\/katex]<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n ![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-22-1024x500.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-23-1024x430.png\")
<\/figure>\n\n\n\n180 Degree Rotation of Vertices<\/strong><\/td><\/tr> [katex]\\begin{aligned}A \\rightarrow A^{\\prime}\\end{aligned}[\/katex]<\/td> [katex] \\begin{aligned}(2, 6) \\rightarrow (-2, -6)\\end{aligned} [\/katex]<\/td><\/tr> [katex] \\begin{aligned}B \\rightarrow B^{\\prime}\\end{aligned} [\/katex]<\/td> [katex] \\begin{aligned}(2, 2) \\rightarrow (-2, -2)\\end{aligned}[\/katex]<\/td><\/tr> [katex] \\begin{aligned}C \\rightarrow C^{\\prime}\\end{aligned} [\/katex]<\/td> [katex] \\begin{aligned}(7, 2) \\rightarrow (-7, -2)\\end{aligned} [\/katex]<\/td><\/tr> [katex] \\begin{aligned}D \\rightarrow D^{\\prime}\\end{aligned}[\/katex]<\/td> [katex] \\begin{aligned}(7, 6) \\rightarrow (-7, -6)\\end{aligned}[\/katex]<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n Problem 1<\/h3>\n\n\n\n
![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-24.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-25.png\")
<\/figure>\n\n\n\n[katex] \\begin{aligned}A \\rightarrow A^{\\prime}\\end{aligned} [\/katex]<\/td> [katex] \\begin{aligned}(1, 4) \\rightarrow (-1, -4)\\end{aligned} [\/katex]<\/td><\/tr> [katex] \\begin{aligned}B \\rightarrow B^{\\prime}\\end{aligned} [\/katex]<\/td> [katex]\\begin{aligned}(5, -2) \\rightarrow (-5, 2)\\end{aligned}[\/katex]<\/td><\/tr> [katex] \\begin{aligned}C \\rightarrow C^{\\prime}\\end{aligned} [\/katex]<\/td> [katex]\\begin{aligned}(-3, -4) \\rightarrow (-4, 3)\\end{aligned}[\/katex]<\/td><\/tr> [katex] \\begin{aligned}D \\rightarrow D^{\\prime}\\end{aligned} [\/katex]<\/td> [katex]\\begin{aligned}(2, -4) \\rightarrow (-2, 4)\\end{aligned}[\/katex]<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n How To Transform a Figure in a 180 Degree Rotation?<\/h2>\n\n\n\n
Problem 2<\/h3>\n\n\n\n
![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-26.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-27.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-28.png\")
<\/figure>\n\n\n\nProblem 3<\/h3>\n\n\n\n
![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-29-1024x728.png\")
<\/figure>\n\n\n\nProblem 4<\/strong><\/h3>\n\n\n\n
![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-30.png\")
<\/figure>\n\n\n\n[katex] \\begin{aligned}\\Delta ABC \\rightarrow \\Delta A^{\\prime}\\Delta B^{\\prime}C^{\\prime}\\end{aligned} [\/katex]<\/td><\/tr> [katex] \\begin{aligned}A \\rightarrow A^{\\prime}\\end{aligned} [\/katex]<\/td> [katex] \\begin{aligned}(1, 0) \\rightarrow (-1, 0)\\end{aligned} [\/katex]<\/td><\/tr> [katex] \\begin{aligned}B \\rightarrow B^{\\prime}\\end{aligned} [\/katex]<\/td> [katex] \\begin{aligned}(5, 4) \\rightarrow (-5, -4)\\end{aligned} [\/katex]<\/td><\/tr> [katex] \\begin{aligned}C \\rightarrow C^{\\prime}\\end{aligned} [\/katex]<\/td> [katex] \\begin{aligned}(8, 0) \\rightarrow (-8, 0)\\end{aligned} [\/katex]<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n ![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-31-1024x569.png\")
<\/figure>\n\n\n\n