given quadrilateral abcd is a parallelogram

The solution 6 is the high school version of solution 2. There is also a more directive version of this task that asks students to use the Pythagorean theorem: 8.G A rectangle in the coordinate plane Students could then consider both of these prerequisite tasks and strategies useful for solving them when presented with this task item as written. Teachers working with 8th grade students may choose to provide scaffolded prerequisite tasks that present the coordinates for four vertices of a quadrilateral with its center at (0,0) in order to employ a single rigid motion (rotation) and then a separate set of coordinates for four vertices forming a quadrilateral with side parallel to the grid lines. Two of the solutions are appropriate for the eighth grade while the other four require knowledge of high school content so ideally students will work on this both in middle school and then later in high school. The open-ended nature of the task makes it ideally suited for extended work and discussion, including sharing of ideas and different approaches. This task is intended for instructional purposes only.

At the same time, it gives an opportunity to relate these new ideas to classical approaches using the Pythagorean Theorem and slopes of lines. This task supports the transformational approach to geometry adapted in the CCSS in a very strong way: the second solution uses rotations and translations in a fundamental way while the fourth solution requires an understanding of similar triangles and consequently of dilations of the plane. One of the advantages to open-ended tasks like this one is that they allow for multiple, creative solutions and, in so doing, show how the different strands of the curriculum are related. Two final solutions have been presented, one making use of the $SSS$ congruence criterion for triangles and the final one using a criterion for when a parallelogram is a rectangle. A second method produces several auxiliary triangles and uses the important notion of similarity.įor this last solution, the teacher may wish to provide extra prompting: drawing theĬoordinate rectangle circumscribing the given quadrilateral is a starting point but some encouragement to study the four triangles left when the given quadrilateral is removed may also be necessary. At the high school level, one way to show that an angle is a right angle is to make algebraic calculations of slopes of the lines containing the two rays of the angle. Also at the eighth grade level, rigid motions of the plane can be applied to the quadrilateral so that its angles are shown to be congruent to angles made by horizontal and vertical grid lines.

One method to achieve this, at the eighth grade level, is with the Pythagorean theorem and its converse. In order to solve this problem, students must show that the four angles in the quadrilateral are all right angles. It is important that different methods of solution be considered because this reveals the connections between different standards. This task has been worded to support a wide range of approaches which help link eighth grade geometry standards with high school geometry (and algebra) standards. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. Students will need ample time and, for some of the

Using this information to show that the four angles are right angles will require some auxiliary constructions. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle.

Given quadrilateral abcd is a parallelogram