Given a segment in the coordinate plane, students find the segments obtained by rotating the given segment by 90° counterclockwise and clockwise about one endpoint.

Students explain the connection between the Pythagorean Theorem and the criterion for perpendicularity.

Students generalize the criterion for perpendicularity of two segments that meet at a point to any two segments in the Cartesian plane.
Students apply the criterion to determine if two segments are perpendicular.

Students state the relationship between previously used formats for equations for lines and the new format a1x + a2y + c, recognizing the segments from (0,0) to (a1, a2) as a normal and - a2/a1 as a slope.
Students solve problems that are dependent upon making such interpretations.

Students recognize parallel and perpendicular lines from slope.
Students create equations for lines satisfying criteria of the kind: “Contains a given point and is parallel/perpendicular to a given line.”

The challenge of programming robot motion along segments parallel or perpendicular to a given segment leads to an analysis of slopes of parallel and perpendicular lines. Students write equations for parallel, perpendicular, and normal lines. Additionally, students will and study the proportionality of segments formed by diagonals of polygons.

Students find the perimeter of a quadrilateral in the coordinate plane given its vertices and edges.
Students find the area of a quadrilateral in the coordinate plane given its vertices and edges by employing Green’s theorem.

Students find the perimeter of a triangle or quadrilateral in the coordinate plane given a description by inequalities.
Students find the area of a triangle or quadrilateral in the coordinate plane given a description by inequalities by employing Green’s theorem.

Students find the perimeter of a triangle in the coordinate plane using the distance formula.
Students state and apply the formula for area of a triangle with vertices (0,0),(x1, y1), and (x2 , y2).

Students find midpoints of segments and points that divide segments into 3, 4, or more proportional, equal parts.

Using coordinates, students prove that the intersection of the medians of a triangle meet at a point that is two-thirds of the way along each median from the intersected vertex.
Using coordinates, students prove the diagonals of a parallelogram bisect one another and meet at the intersection of the segments joining the midpoints of opposite sides.

Students name several points on a line given by a parametric equation and provide the point-slope equation for a line given by a parametric equation.
Students determine whether lines given parametrically are parallel or perpendicular.

Using observations from a pushing puzzle, explore the converse of Thales' theorem: If triangle ABC is a right triangle, then A, B, and C are three distinct points on a circle with segment AC a diameter.
Prove the statement of Thales' theorem: If A, B, and C are three different points on a circle with segment AC a diameter, then angle ABC is a right angle.

Identify the relationships between the diameters of a circle and other chords of the circle.

Inscribe a rectangle in a circle.
Understand the symmetries of inscribed rectangles across a diameter.

Explore the relationship between inscribed angles and central angles and their intercepted arcs.

Prove the inscribed angle theorem: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle.
Recognize and use different cases of the inscribed angle theorem embedded in diagrams. This includes recognizing and using the result that inscribed angles that intersect the same arc are equal in measure.

Define the angle measure of arcs, and understand that arcs of equal angle measure are similar.
Restate and understand the inscribed angle theorem in terms of arcs: The measure of an inscribed angle is half the angle measure of its intercepted arc.
Explain and understand that all circles are similar.

Congruent chords have congruent arcs, and the converse is true.
Arcs between parallel chords are congruent.