![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-7.png\")
<\/figure>\n\n\n\nIn a coordinate plane, when the point [katex](x, y)[\/katex] is rotated in a 90-degree clockwise direction, the projected image will have a coordinate of [katex](y, -x)[\/katex]. This also means that when a figure is rotated in the same direction on a Cartesian plane, all the points will exhibit the same behavior.<\/p>\n\n\n\n To better understand how the 90 degree clockwise rotation works, let\u2019s take a look at the rotations for the following figures: [katex]A \\rightarrow A^{\\prime}[\/katex] and [katex]\\Delta ABC \\rightarrow A^{\\prime}B^{\\prime} C^{\\prime} [\/katex]. When working on rotations, always refer to the fixed point when turning the pre-image (the original figure) to form the resulting image of the transformation.<\/p>\n\n\n\n First, focus on how the point, [katex]A[\/katex], is rotated in a 90 degree clockwise direction with the origin placed as its reference point. Construct two rays by connecting the origin to the pre-image and the resulting image. We can see that the two rays, [katex]\\overrightarrow{OA}[\/katex] and [katex]\\overrightarrow{OA^{\\prime}}[\/katex], form a right angle. This confirms that point A was rotated about 90o in a clockwise direction to reach the projected point, [katex]A^{\\prime}[\/katex].<\/p>\n\n\n\n Now, focus on the transformation of [katex]\\Delta ABC[\/katex] and observe the angles formed by the rays from the following three pairs of points: [katex]A \\rightarrow A^{\\prime}[\/katex], [katex]B \\rightarrow B^{\\prime}[\/katex], and [katex]C\\rightarrow C^{\\prime}[\/katex].<\/p>\n\n\n\n All three rotations are in a clockwise direction, so we can confirm that [katex]\\Delta A^{\\prime}B^{\\prime}C^{\\prime}[\/katex] is the result when [katex]\\Delta ABC[\/katex] is rotated at a right angle and in a clockwise direction. Now, it\u2019s time to observe how the significant points\u2019 coordinates behave after rotating the two figures.<\/p>\n\n\n\n These transformations on the core points confirm that when rotating a point or a figure in a 90 degree clockwise direction, the x and y-coordinates switch their places, where the new y-coordinate is the negative equivalent of the pre-image\u2019s x-coordinate – [katex]P: (x, y) \\rightarrow P^{\\prime}: (y, -x)[\/katex].<\/p>\n\n\n\n The four points and their corresponding rotations are graphed on the Cartesian plane shown below. <\/p>\n\n\n\n Which of the following points do not reflect a 90-degree clockwise rotation?<\/p>\n\n\n\n A. [katex]A\\rightarrow A^{\\prime}[\/katex]<\/p>\n\n\n\n B. [katex]B\\rightarrow B^{\\prime}[\/katex]<\/p>\n\n\n\n C. [katex]C\\rightarrow C^{\\prime}[\/katex]<\/p>\n\n\n\n D. [katex]E\\rightarrow E^{\\prime}[\/katex]<\/p>\n\n\n\n There are two ways to confirm the rotation performed on each of the pair of pre-image and image points. We can do this by measuring the angle formed by the rays connecting the origin and pairs of points. <\/p>\n\n\n\n By quickly scanning the angles formed by the four pairs of points, we can see that they all form angles close to 90o<\/sup>. This is an approximation since we\u2019re just inspecting the angles. To confirm that these angles measure 90o<\/sup>, use a compass and see if the angle formed by each pair of points form a right angle. Now, observe the direction of the rotation.<\/p>\n\n\n\n This means that the rotation, [katex]E \\rightarrow E^{\\prime}[\/katex], does not exhibit the 90 degree clockwise rotation but instead, it is the result of rotating [katex]E[\/katex] in a 90-degree counterclockwise direction and still with respect to the origin. <\/p>\n\n\n\n A more accurate way of confirming the rotations of this point is by observing their x and y-coordinates.<\/p>\n\n\n\n By observing the effect of the rotation on each of the resulting images of the four points, we can see that the first three rotations exhibit what we expect from a 90 degree clockwise rotation: [katex](x, y) \\rightarrow (y, -x)[\/katex]. Meanwhile, for the fourth point, [katex](5, 2) \\rightarrow (-2, 5)[\/katex], the coordinates do not exhibit this behavior. Instead, this is how we expect the coordinates of a point will behave when turned 90 degrees in a counterclockwise rotation. Hence, the correct answer is D: [katex]E\\rightarrow E^{\\prime}[\/katex].<\/p>\n\n\n\n Now, what if we\u2019re given the figure to rotate in a 90-degree clockwise direction? We\u2019ll show you how in the next section!<\/p>\n\n\n\n There are two ways to transform a figure in a 90 degree clockwise rotation: 1) use a compass and a straightedge to accurately project the image of a figure and 2) project the x and y-coordinates of the figure when it is on a xy-plane: [katex](x, y) \\rightarrow (y, -x)[\/katex].<\/p>\n\n\n\n To decide which method would best work for a given problem, observe what\u2019s given. If the figure being rotated is outside a Cartesian plane, use the first method to accurately graph the projected image. Now, if we\u2019re given a figure graphed on a Cartesian plane, the second method is a much simpler approach.<\/p>\n\n\n\n In the next problems, we\u2019ll show different examples of applying the second method. When ready, try to work on the problems we\u2019ve prepared just for you!<\/p>\n\n\n\n Predict the result when the pre-image (the pizza) is rotated in a 90-degree clockwise direction with respect to the reference point as shown below.<\/p>\n\n\n\n The goal here is to have an idea of the resulting position of the image after being rotated 90 degrees in a clockwise direction. To do this, construct the first ray by connecting the reference point to the object.<\/p>\n\n\n\n Construct the second ray of the angle by creating a right angle (angle measuring 90o), so the two line segments must form an \u201cL\u201d-shape and is going to the right of the first ray. We do this so that the clockwise direction is accounted for.<\/p>\n\n\n\n Project the pre-image of the pizza and follow the direction as a guide in rotating the pizza in a 90 degree clockwise rotation. Observe how the pizza rotated to better understand how images are formed after being rotated 90 degrees clockwise.<\/p>\n\n\n\n What are the vertices of the projected image after [katex]\\Delta ABC[\/katex] is rotated 90 degrees and in a clockwise direction? Extend the xy-plane shown below then graph the rotated image of [katex]\\Delta ABC[\/katex].<\/p>\n\n\n\n Now that the image we\u2019re rotating is graphed on a xy-plane, take note of the figure\u2019s key points first. For [katex]\\Delta ABC[\/katex], its key points are its vertices: [katex]A= (2, 5)[\/katex], [katex]B= (1, 1)[\/katex], and [katex]A= (7, 1)[\/katex]. To reflect the triangle, reflect these three vertices first by applying the rule for 90 degree clockwise rotation: [katex](x, y) \\rightarrow (y, -x)[\/katex].<\/p>\n\n\n\n The table summarizes the new vertices of our triangle. Now, let\u2019s plot the three rotated vertices on the xy-plane then connect them to form the image of the triangle, [katex]A^{\\prime}B^{\\prime}C^{\\prime}[\/katex].<\/p>\n\n\n\n The 90-degree clockwise rotation is a special type of rotation that turns the point or a graph a quarter to the right. In this article, we\u2019ll show you how easy it is to perform this rotation and show you techniques to remember to master rotating figures in a 90-degree clockwise 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-415","post","type-post","status-publish","format-standard","hentry","category-geometry"],"_links":{"self":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/415","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=415"}],"version-history":[{"count":8,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/415\/revisions"}],"predecessor-version":[{"id":438,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/415\/revisions\/438"}],"wp:attachment":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/media?parent=415"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/categories?post=415"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/tags?post=415"}],"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}(y, -x)\\end{aligned}[\/katex]<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n ![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-8.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-9.png\")
<\/figure>\n\n\n\n90 Degree Clockwise Direction<\/strong><\/td><\/tr> [katex] \\begin{aligned}A \\rightarrow A^{\\prime}\\end{aligned}[\/katex]<\/td> [katex]\\begin{aligned}(2, 4) \\rightarrow (4, -2)\\end{aligned}[\/katex]<\/td><\/tr> [katex] \\begin{aligned}\\Delta ABC \\rightarrow \\Delta A^{\\prime}B^{\\prime}C^{\\prime}\\end{aligned}[\/katex]<\/td> [katex] \\begin{aligned}A: (2, 4) \\rightarrow (4, -2)\\\\B: (5, 6) \\rightarrow (6, -5)\\\\C: (4, 2) \\rightarrow (2, -4)\\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-10.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-11.png\")
<\/figure>\n\n\n\n[katex]\\begin{aligned}A \\rightarrow A^{\\prime}\\end{aligned}[\/katex]<\/td> [katex]\\begin{aligned}(2, 3) \\rightarrow (3, -2)\\end{aligned}[\/katex]<\/td><\/tr> [katex]\\begin{aligned}B \\rightarrow A^{\\prime}\\end{aligned}[\/katex]<\/td> [katex]\\begin{aligned}(1, -4) \\rightarrow (-4, -1)\\end{aligned}[\/katex]<\/td><\/tr> [katex]\\begin{aligned}C \\rightarrow C^{\\prime}\\end{aligned}[\/katex]<\/td> [katex]\\begin{aligned}(2, 3) \\rightarrow (3, -2)\\end{aligned}[\/katex]<\/td><\/tr> [katex]\\begin{aligned}E\\rightarrow E^{\\prime}\\end{aligned}[\/katex]<\/td> [katex]\\begin{aligned}(5, 2) \\rightarrow (-2, 5)\\end{aligned}[\/katex]<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n How To Transform a Figure in a 90 Degree Clockwise Rotation?<\/h2>\n\n\n\n
Problem 2<\/h3>\n\n\n\n
![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-17-1024x459.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-13.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-14.png\")
<\/figure>\n\n\n\nProblem 3<\/h3>\n\n\n\n
![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-15.png\")
<\/figure>\n\n\n\n[katex]\\begin{aligned}\\boldsymbol{(x, y) \\rightarrow (y, -x)} \\end{aligned}[\/katex]<\/td><\/tr> [katex]\\begin{aligned}A\\rightarrow A^{\\prime}\\end{aligned}[\/katex]<\/td> [katex]\\begin{aligned}(2, 5) \\rightarrow (5,-2)\\end{aligned}[\/katex]<\/td><\/tr> [katex]\\begin{aligned}B\\rightarrow B^{\\prime}\\end{aligned}[\/katex]<\/td> [katex]\\begin{aligned}(1, 1) \\rightarrow (1,-1)\\end{aligned}[\/katex]<\/td><\/tr> [katex]\\begin{aligned}C\\rightarrow C^{\\prime}\\end{aligned}[\/katex]<\/td> [katex]\\begin{aligned}(7, 1) \\rightarrow (1,-7)\\end{aligned}[\/katex]<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n ![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/02\/image-16.png\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"