G-GPE. Expressing Geometric Properties with Equations

G-GPE.A. Translate between the geometric description and the equation for a conic section

G-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.

• Explaining the equation for a circle
• Slopes and Circles

G-GPE.A.2. Derive the equation of a parabola given a focus and directrix.

• Defining Parabolas Geometrically

G-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.

• No tasks yet illustrate this standard.

G-GPE.B. Use coordinates to prove simple geometric theorems algebraically

• Coordinates of equilateral triangles
• Coordinates of Points on a Circle
• Is this a rectangle?

G-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 $(1, \sqrt{3})$ lies on the circle centered at the origin and containing the point $(0, 2)$.

• A Midpoint Miracle
• Unit Squares and Triangles

G-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).

• A Midpoint Miracle
• Equal Area Triangles on the Same Base I
• Equal Area Triangles on the Same Base II
• Parallel Lines in the Coordinate Plane
• Slope Criterion for Perpendicular Lines
• Triangles inscribed in a circle
• Unit Squares and Triangles
• When are two lines perpendicular?

G-GPE.B.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

• Finding triangle coordinates
• Scaling a Triangle in the Coordinate Plane

G-GPE.B.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

• Squares on a coordinate grid
• Triangle Perimeters