Plane math1/20/2024 ![]() Because they are so simple, it is hard to give precise definitions of them, so instead we aim to. The image of $g$ is $\triangle A^$ on the other, these triangles are not similar and so their angles are not congruent. The simplest objects in plane geometry are points and lines. The image of $f$ is $\triangle A^\prime B^\prime C^\prime$ which is congruent to the original triangle but it is displaced to the left by 3 units. If you use a protractor to measure the angle of a triangle, youre doing. ![]() The teacher should prompt students for a geometric description of what $f$ and $g$ do so that students understand what transformations of the plane these functions represent.īelow is a picture of the images of the triangle after applying the maps $f$ and $g$: In math, plane geometry is a system of measuring flat, two-dimensional shapes. ![]() The task does not specify that $f$ is a translation to the left by 3 units and that $g$ is a horizontal stretch by a factor of 3. In the solution a similarity argument is provided but students familiar with trigonometry could also calculate the angles in the different triangles and verify in this way that horizontal stretch changes the angles of this particular triangle. The fact that horizontal stretch does not preserve angles is visibly clear but requires more careful thinking. A plane, as defined by Euclid, is a surface which lies evenly with the straight lines on itself. The only line segments whose distance is preserved by this horizontal stretch are vertical lines and the teacher may wish to have students investigate this as a follow up question. The fact that horizontal stretch does not preserve distances can be seen from the pictures or using the Pythagorean theorem. The generalization of the plane to higher dimensions is called a hyperplane. ![]() The goal of this task is to compare a transformation of the plane (translation) which preserves distances and angles to a transformation of the plane (horizontal stretch) which does not preserve either distances or angles. A plane is a two-dimensional doubly ruled surface spanned by two linearly independent vectors. ![]()
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