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# High School: Geometry: Expressing Geometric Properties with Equations

### 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.

### HSG.GPE.A.2

Derive the equation of a parabola given a focus and directrix.

### 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.

### HSG.GPE.B.4

Use coordinates to prove simple geometric theorems algebraically.

### HSG.GPE.B.5

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.

### HSG.GPE.B.6

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

### HSG.GPE.B.7

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.