8.G Cutting a rectangle into two congruent triangles
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1 8.G Cutting a rectangle into two congruent triangles Task ABCD AC Below is a picture of rectangle with diagonal. ACD 180 D A A C C a. Draw the image of triangle when it is rotated about vertex. Call the image of point under the rotation and the image of point. b. Explain why and why is parallel to. DA DA DC A C D c. Show that can be translated to. Conclude that is AB
2 congruent to. d. Show that is congruent to with a sequence of translations, rotations, and/or reflections different from those chosen in parts (a) and (c). IM Commentary This task shows the congruence of two triangles in a particular geometric context arising by cutting a rectangle in half along the diagonal. The teacher may wish to have students think about part (d) of the problem first before going through parts (a) through (c). The grid has been added to the background as a geometric aid and as a means of determining lengths of sides. For part (d), the grid may need to be extended depending on which translations, rotations, and reflections the students choose to use. A good way of reasoning about this problem abstractly would be to observe that the triangles are both right triangles and the lengths of the sides that come together to form the right angles are 4 and 8 units respectively. So to show that they are congruent we just need to "align" the right angles and the sides with corresponding lengths. Some questions which teachers may wish to have students explore further are: Is it possible to show the congruence using only translations? Is it possible to show the congruence with a reflection and then a translation? Is it possible to show the congruence using only reflections? The task is primarily intended for instructional purposes. Part (b) has been added explicitly to help students visualize how certain 180 rotations influence a coordinate grid. Also part (d) has many possible answers, providing an ideal opportunity to have students share their (different) constructions with one another. If the teacher wishes to give students a chance to be creative and explore many different ways to transform to then part (d) alone can be used. Solution 180 D D a. Rotation by about takes any line through to itself but switches the orientation of the line. So, for example, line is taken to but the notions of up DA DA
3 A D A D D C D 180 D C 8 D C 180 D 8 D and down on this vertical line are reversed. So point, which is four units below will be mapped to the point which is four units above. The points and, on the other hand, lie on a horizontal line through. So rotation by about has the effect of switching left and right on this line. Since lies units to the right of, its image,, after rotating about will be the units to the left of. This is pictured below: 180 D D b. As explained in part (a), rotation by about maps any line through to itself, D exchanging the two rays through which make up the line. This means that DA = DA since DA is the rotation of line DA by 180 about D. Similarly, DC = DC. So DC is a horizontal line of the grid, as is AB, and so these two lines are parallel, remaining four units apart from one another. D B C A A C c. To send to we need to translate by units to the right and units down. Translating by units to the right and units down also takes to and to as is seen in the picture below:
4 Since the vertices of triangle map to the vertices of triangle and translations take line segments to line segments, this means that translation units to the right and units down takes triangle to triangle. The combination of a rotation about followed by a translation units to the right and four units down establishes the congruence of triangles and. d. One of the most interesting ways to show the congruence of and is to rotate by about the intersection point of the diagonals and of rectangle boxes to the right of, labelled A C D 4 A C D 180 D 8 A C D CAB CAB CAB 180 AC BD ABCD A O in the diagram below:. Using the coordinate grid, this is the point two boxes up and four 8
5 180 O To see why rotation by about maps to, note that we know from parts (a) and (b) that rotation by 180 will map line AC to itself (and similarly for line which also passes through ). Segments and are congruent: each is DB O OC OA O the hypotenuse of a right triangle with legs of length and and so each has length by the Pythagorean theorem. Since rotation by fixes and interchanges the two rays OA and OC this means that this rotations maps A to C and C to A. Similar reasoning applied to the rays and shows that rotation by maps to OD OB 180 B D B D 180 O ACD CAB and to. Hence rotation by about maps triangle to triangle and shows that they are congruent. PQD There are many other ways to show the congruence of and. For example, we could use two reflections to move to the position of from part (a): first a reflection about line DC and then a reflection about AD (or first a reflection about AD and then a reflection about DC ). This particular method gives an example of two reflections which commute, that is the order in which they are applied does not influence the resulting transformation of the plane.
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