Mathematics | Course : Model Mathematics II (Integrated Pathway)
Domain - Expressing Geometric Properties with Equations
Cluster - Translate between the geometric description and the equation for a conic section.
[MII.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.
- Pythagorean Theorem
For any right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse.
[MI.A-REI.D.10] -
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Show that any point on the graph of an equation in two variables is a solution to the equation.
[MII.A-REI.B.4.a] -
Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
[MII.A-REI.C.7] -
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
[MII.G-GPE.A.2] -
Derive the equation of a parabola given a focus and directrix.
[MII.G-GPE.B.4] -
Use coordinates to prove simple geometric theorems algebraically including the distance formula and its relationship to the Pythagorean Theorem. 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).