HS Math Concept: Expressing Geometric Properties with Equations

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Expressing Geometric Properties with Equations

Resources

Translate between the geometric description and the equation for a conic section.
HSG-GPE.A.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. tools-icon
HSG-GPE.A.2. Derive the equation of a parabola given a focus and directrix. tools-icon
HSG-GPE.A.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.
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3.1 Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, use the method for completing the square to put the equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola and graph the equation. CA  tools-icon
Use coordinates to prove simple geometric theorems algebraically.
HSG-GPE.B.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).  tools-icon
HSG-GPE.B.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).  tools-icon
HSG-GPE.B.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
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HSG-GPE.B.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.  tools-icon