G.GPE

Expressing Geometric Properties with Equations

G.GPE.1(9/10) Write the equation of a circle given the center and radius or a graph of the circle; use the center and radius to graph the circle in the coordinate plane.G.GPE.2(+) Derive the equation of a circle of given center and radius using the Pythagorean Theorem; graph the circle in the coordinate plane;G.GPE.3(+) Complete the square to find the center and radius of a circle given by an equation.G.GPE.4(+) Derive the equation of a parabola given a focus and directrix; graph the parabola in the coordinate plane.G.GPE.5(+) 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; graph the ellipse or hyperbola in the coordinate plane.G.GPE.6(9/10) Use coordinates to prove simple geometric theorems algebraically, including the use of slope, distance, and midpoint formulas For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle.G.GPE.7(9/10) 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.8(9/10) Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, including the use of the distance and midpoint formulas.
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