Lines are taken to lines, and line segments to line segments of the same length. Angles are taken to angles of the same measure. Parallel lines are taken to parallel lines. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Focus Standards of Mathematical Practice:. Instructional Outcomes:. Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:.
For example, arrange three copies of the same triangle so that the three angles appear to form a line. Give an argument in terms of transversals why this is so. Enduring Understandings:. If the scale factor of a dilation is equal to 1, the image resulting from the dilation is congruent to the original figure.
Essential Questions:. Unit 4 The Number System Grade 8 Math Unit Length and Description: 30 days In this unit, students analyze two- and three-dimensional space and figures using congruence and similarity. Standards: 8.
Standards Clarification: The skill of transforming geometric items as well as the properties of transforming these items will extend to develop and establish the criteria for figure congruence and figure similarity.
This standard does not include the transformation of figures. Standards Clarification: Translations, reflections, and rotations are called rigid transformations because they do not change the size or shape of an item.
Characteristics such as the length of line segments, angle measures, and parallel lines are unchanged by these three types of transformations. Standards Clarification: Because size and shape are preserved under translations, reflections, and rotations, the result of these transformations is an exact copy of the original figure.
When two figures have the exact same size and shape, they are called congruent figures. In essence, we will be looking at the "slope" of each line segment involved. Part 1: Given point C -2,-1 , center of dilation of -4,-9 , and scale factor of 2, find C'. By observation, point C is 8 vertical units above the center of dilation.
Under a scale factor of 2, point C' needs to be 16 vertical units from the center. Also, point C is 2 horizontal units right of the center of dilation. Point C' needs to be 4 horizontal units right of the center.
Starting at the center of dilation -4,-9 , move 16 units up and 4 units to the right to find C' at 0,7. A -4,-6 , B 3,-6 , and C -2, By observing vertical and horizontal distances from the center of dilation, as seen in Part 1, you can find the remaining two coordinates of the dilated triangle. The counting of vertical and horizontal distances shown above is a simple and easy way to find the coordinates for a dilation not centered at the origin.
FYI: Another Method A dilation not centered at the origin, can also be thought of as a series of translations, and expressed as a formula. Translate the center of the dilation to the origin, apply the dilation factor as shown in the "center at origin" formula, then translate the center back undo the translation.
Subtracting the coordinate values of the center of dilation will move the center to the origin. Given center of dilation at a,b , translate the center to 0,0 : x - a , y - b.
Write a coordinate rule to find the vertices of a dilation with center 4,-2 and scale factor of 3. Let x,y be a vertex of the figure. Dilations MathBitsNotebook. For an intuitive review of dilations, see the Refresher section Transformations: Dilations. Now, let's expand that knowledge of dilations in relation to geometry.
A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. The description of a dilation includes the scale factor constant of dilation and the center of the dilation.
The center of a dilation is a fixed point in the plane about which all points are expanded or contracted. Dilations create similar figures!
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