Standard - G-GPE.1:
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Standard - G-GPE.2:
Derive the equation of a parabola given a focus and directrix.

Standard - G-GPE.3:
Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Standard - G-GPE.4:
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

Standard - G-GPE.5:
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Standard - G-GPE.6:
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Standard - G-GPE.7:
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.