G-GPE: Geometry: Expressing Geometric Properties with Equations (G-GPE)

Geometry: Expressing Geometric Properties with Equations (G-GPE)

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G-GPE.1Derive 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.2Derive the equation of a parabola given a focus and directrix.G-GPE.3Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.G-GPE.4Use 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.5Prove 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.6Find the point on a directed line segment between two given points that partitions the segment in a given ratio.G-GPE.7Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
Example Problems
The equation of a circle is given below.


What is its center?
Write the equation for a parabola with a focus at and a directrix at .
What are the foci of the hyperbola represented by the equation
?
Triangle ABC has the following vertices:
A

B

C

Is triangle ABC a
right triangle?
Write the equation of a line that is parallel to and that passes through the point .
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