[G.GPE.] Expressing Geometric Properties With Equations

[G.GPE.] Expressing Geometric Properties with Equations

Translate between the geometric description and the equation for a conic section.

• [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.

  • [G.GPE.2] Derive the equation of a parabola given a focus and directrix.

Use coordinates to prove simple geometric theorems algebraically. [Include distance formula; relate to Pythagorean Theorem.]

• [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).

• [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).

• [G.GPE.6] Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

  • [G.GPE.7] (*) Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.