Investigation: Congruent Figures using Transformations

Transcription

1 Investigation: ongruent Figures using Transformations In middle school, you were introduced to concepts about congruence. You learned that a figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. You described sequences that showed the congruence between two congruent figures. ongruence should be familiar to you, but you will now study it with more precision and focus on the language you use to discuss it. 1. Explore the transformation(s) below and describe it as precisely as you can. Recall from middle school that a correspondence between two triangles is a pairing of each vertex of one triangle with one and only one vertex of the other triangle. When reasoning about figures, it is useful to be able to refer to corresponding parts of the two figures. rigid motion always produces a one-to-one correspondence between the points in a figure (the pre-image) and points in its image. rigid motion also maps each part of the figure to a corresponding part of the image. 2. omplete the following correspondences of the parts for the pre-image PQR to the image XYZ (from above). orresponding ngles PQR QRP RPQ orresponding Sides PQ QR RP i. What can you determine about the corresponding parts of PQR and XYZ? ii. Write statements similar to those in the table above to demonstrate the relationship between the corresponding parts. iii. What can you determine about PQR and XYZ? by. Hollifield for uncombe ounty Schools

2 3. student, Phillip, explored the same composition of transformations and wrote the following to prove that the image XYZ was congruent to its pre-image PQR. Parts of his justification have been left out. What details do you think he included in these blanks? To map PQR to XYZ, we first. Then,. Finally, the second image to obtain XYZ. Since each transformation is a rigid motion and preserves distance and angle measure, PQR XYZ. Precision in language and details are important components of creating mathematical arguments. When completing the following exercises focus on your precision in language and the specificity of the details you provide as you prove or write a convincing argument that the figures are or are not congruent. 4. is a square, and is one diagonal of the square. is a reflection of across segment. omplete the table below, identifying the missing corresponding angles and sides. orresponding ngles orresponding Sides i. re the corresponding sides and angles congruent? Justify your response. ii. Is? Write a proof/convincing argument using transformational geometry to justify your response. 5. The figure below represents a rotation of E 80 clockwise around vertex E. Name the figure formed by the image of E. Name all corresponding angles and sides. re the figures congruent? Write a proof/convincing argument using transformational geometry to justify your response. R G O orresponding ngles orresponding Sides E M by. Hollifield for uncombe ounty Schools

3 6. In the figure below, the triangle on the left has been mapped to the one on the right. Identify all six pairs of corresponding parts (vertices and sides). What rigid motion mapped the triangle on the left to the one on the right? Write a proof/convincing argument using transformational geometry to justify your response. T Q L 7. Prove GRE utilizing transformational geometry. What should be included in the proof for the argument to be valid and justified? E G R M 8. ssume that the following figures are drawn to scale. Use your understanding of congruence to explain why square and rhombus GHIJ are not congruent. 9. G H J I by. Hollifield for uncombe ounty Schools

4 Problem Set 1. Use your understanding of congruence to explain the following: a. Why can t a triangle be congruent to a quadrilateral? b. Why can t an isosceles triangle be congruent to a triangle that is not isosceles? 1.. What rigid motion(s) maps onto? Find two possible solutions. 2. ll of the smaller triangles are congruent to each other in the figure to the right. Which of the following rigid motions will map Z onto Z? If the transformation does map Z onto Z, write the correspondence statements for the sides and angles. Z X Rigid Motion Maps Z onto? Z orrespondence Statements Y Translation Reflection across ZX 180 rotation about vertex 180 rotation about the midpoint of ZY followed by a 180 rotation about the midpoint of ZX by. Hollifield for uncombe ounty Schools

5 The next two tasks show a sequence of rigid motions that map a pre-image onto a final image. Identify each rigid motion in the sequence. heck the congruence of each set of corresponding sides and angles through all steps in the sequence, proving that the pre-image is congruent to the final image by showing that every side and every angle in the pre-image maps to its corresponding side and angle in the image. Record the sequence of rigid motions and the relationship for the corresponding parts in the table. Finally, make a statement about the congruence of the pre-image and final image. 3. Sequence of Rigid Motions orresponding Sides orresponding ngles Triangle ongruence Statement 4. Sequence of Rigid Motions orresponding Sides orresponding ngles Triangle ongruence Statement 5. Prove the following figures are congruent utilizing transformational geometry. What is required for this proof/mathematical argument to be valid and justified? by. Hollifield for uncombe ounty Schools

6 6. Given two triangles, one with vertices,, and, and the other with vertices X, Y, and Z, there are six different correspondences of the first with the second. a. One such correspondence is the following: Write the other five correspondences. Z X Y b. If all six of these correspondences show congruence, then what can you say about (and XYZ)? c. If two of the correspondences show congruence, but the others do not, then what can you say about (and XYZ)? d. Why can there be no two triangles where three of the correspondences show congruence, but the others do not? by. Hollifield for uncombe ounty Schools