G-GPE: Expressing Geometric Properties with Equations

Expressing Geometric Properties with Equations

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Expressing Geometric Properties with Equations uses coordinates to prove geometric theorems and derive equations of conic sections. Circle equations are derived from the Pythagorean theorem, and parabola equations are derived from the focus-directrix definition. Slope, distance, and midpoint formulas serve as tools for proving properties of triangles, quadrilaterals, and polygons in the coordinate plane.

G-GPE.1Determine or 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.2Determine or derive the equation of a parabola given a focus and directrix.G-GPE.3(+) Derive the equations of ellipses and hyperbolas given foci and directrices.G-GPE.4Perform simple coordinate proofs. 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, root 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?
The parabola is reflected across the -axis.

What is the equation of the new parabola?
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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