Each numerical value on the watch measures 5 minutes for the minute hand and 1 hour for the hour hand. Since the pivot point is at the number 8 for the watch, the reading of the minute hand is 40 minutes. Since the rotation is clockwise, the hour hand is also rotated clockwise to represent an hour after 12:10, the correct time after turning the minute hand is 12:40. If we rotate a number of 270 degrees clockwise, each point in the given figure must be changed from (x, y) to (-y, x) and graphically represent the rotated figure. A rotation is an isometric transformation that rotates each point in a figure at a specific angle and direction around a fixed point. As you can see, both of our experiments follow these rules. Let`s start by finding the coordinates of the vertices of our original pentagon. The rule for 90° counterclockwise is (x,y) becomes (-y,x), we apply the rule to find the vertices of our new pentagon. Let`s take a closer look at the two rotations of our experiment.
In our first experiment, when we turn point A (5.6) 90° clockwise around the origin to produce point A` (6.-5), the y value of point A becomes the x value of point A` and the x value of point A becomes the y value of point A`, but with the opposite sign. The coordinates of the vertices of the triangle ABC, which can be represented graphically in the coordinate plane, are (A(-8,-6)), (B(-2,-6)) and (C(-5,-3)). The triangle is rotated 90° clockwise around the origin to create the triangle (A`B`C`). Which of the following vertices is the triangle (A`B`C`)? One last problem of practice. Trapezoidal PQRS, where P (-3,-5), Q (3,-5), R (5,-2) and S (-5,-2) are rotated 90° clockwise around the origin to create trapezoidal P`Q`R`. Create the two trapezoids in the coordinate grid. In fact, the angle of rotation is twice as high as the acute angle formed between the intersecting lines. Rotations are everywhere you look. Soil is the most common example that revolves around an axis. The wheel of a car or bicycle rotates around the central bolt. These two examples can be rotated 360°. There are other forms of rotation that are less than a full 360° rotation, such as a character or object shot in a video game.
Formally, a rotation is a form of transformation that rotates a figure around a point. We call this point the center of rotation. A figure and its rotation retain the same shape and size, but point in a different direction. A figure can be rotated clockwise or counterclockwise. Another great example of rotation in real life is a Ferris wheel, where the center hub is the center of rotation. Let`s look at another problem. The Pentagon`s QRSTU appears in the coordinate grid. Rotate the QRSTU Pentagon 90° counterclockwise to create the Q`R`S`T`U` PENTAGON. Now that we know how to rotate a point, let`s look at the rotation of a figure in the coordinate grid. To rotate the triangle ABC around the origin 90° clockwise, we would follow the rule (x,y) → (y,-x), where the y value of the origin point becomes the new x value and the x value of the origin point becomes the new y value with the opposite sign. Let`s apply the ruler to the vertices to create the new triangle A`B`C`: If you rotate a point with 180° coordinates with (x,y) coordinates around the origin counterclockwise or clockwise, a point with the coordinates ((-x,-y) is created. If we insert the coordinates of the point (A) into our formula to find the rotated point, we get: Here, the triangle is rotated 270° clockwise.
The rule that we have to apply here is An excellent mathematical tool that we use to display rotations is the coordinate grid. Let`s start by rotating a point around the center point (0.0). If you take a coordinate grid and draw a point, then rotate the paper 90° or 180° clockwise or counterclockwise around the origin, you can find the location of the rotated point. Let`s look at a real example, here we drew point A at (5.6), and then we turned the paper 90° clockwise to create point A`, which is at (6,-5). Let`s take a look at another rotation. Let us rotate the triangle ABC 180° around the origin counterclockwise, although rotating a 180° number clockwise and counterclockwise uses the same rule, namely (x,y) is (-x,-y), where the coordinates of the vertices of the rotated triangle are the coordinates of the original triangle with the opposite sign. Let`s apply the rule to the vertices to create the new triangle A`B`C`: A closer look at the coordinates of the vertices shows that the coordinates of K`L`M`N` are the same as the vertices of the original kite, but with the opposite sign. Let`s look at the rules, the only rule where the values of x and y do not change, but their sign changes, is the 180° rotation. F(-4, -2), G(-2, -2) and H(-3, 1) are the three vertices of a triangle. If this triangle is rotated 270° clockwise, locate the vertices of the rotated figure and graph. Here is four-sided ABCD. To rotate the four-sided ABCD by 90° counterclockwise around the origin, we use the rule (x,y) becomes (-y,x).
Let`s apply the rules to vertices to create A`B`C`D with four sides: Fortunately, these experiments allowed mathematicians to develop rules for the most common rotations on a coordinate grid, assuming the origin (0,0) as the center of rotation. Here are the rotation rules: In terms of coordinates, point A (3,-4)) is rotated 180° counterclockwise around the origin to create the pivoted point (A`). Which of the following is the ordered pair for (A`)? The most common rotations are rotations of 180° or 90° and sometimes rotations of 270° around the origin and affect each point of a figure as follows: The clock reads 12:10. If the minute hand rotates 180° clockwise around the origin, what time will it be? And just as we have seen how two consecutive reflections on parallel lines correspond to a translation, when a figure is reflected twice on intersecting lines, this composition of reflections is equal to one rotation. The ruler below can be used to rotate 270 degrees clockwise. In our second experiment, point A (5.6) is rotated 180° counterclockwise around the origin to produce A` (-5,-6), where the x and y values are the same as point A, but with opposite signs. We start by deciding which ruler to use for a 90° clockwise rotation around the origin. We will use (x,y) to (y,-x). Now let`s apply the rule to the coordinates of PQRS vertices. If you rotate a point with the coordinates ((x,y)) clockwise by 90°, you get a point with the coordinates ((y,-x)). If we replace the coordinates of the point where water enters a paddle, we get our rotation point of ((9,-3)).
Thus, the coordinates of the point where water is released from the water wheel are ((9,-3)). Now I want you to try some practice problems yourself. Kite KLMN is displayed in the coordinate grid. The kite was originally rotated around to create the dragon K`L`M`N`. Can you determine which KLMN Kite rotation created K`L`M`N`? Finally, a figure in a plane has rotational symmetry if the figure can be mapped to itself by rotating 180° or less. This means that if we rotate an object by 180° or less, the new image looks exactly like the original model.
