Mathematics Common Core Standards: Geometric Properties as Equations
Students use what they know about operations in algebra to demonstrate (or prove) certain aspects or characteristics of geometric shapes for Common Core Standards. For example, if you know that the three interior angles of a triangle must add up to 180 degrees and that the first two angles are 70 and 50 degrees, you know that the third angle is 180 – 70 – 50 = 60 degrees.
Students also begin to explore conic sections (or simply conics) — curves, circles, or ellipses formed by a plane slicing through a cone.
Upon graduation, students need to be able to translate between the equation and the graphical representation of conic sections.
If the center of a circle is represented by (h, k) as an ordered pair, then the equation of a circle is (x – h)2 + (y – k)2 = r2, with r being the radius. For our purposes, let’s say that the center of the circle is at (3, 4) and that the radius of the circle is 5.
If you graph this on a coordinate plane, then all points lying on the circumference of the circle can substitute for the values of x and y in the equation.
To test whether you’ve graphed the circle correctly, pick a point that you know should be on the circumference and insert it into the equation. For example, the ordered pair (8, 4) should lie on the circumference. When you substitute (8, 4) for x and y, the equation is still true.
(8 – 3)2 + (4 – 4)2 = 25
52 = 25
The equation represents a cross section of a cone taken parallel to the base. The equation defines a circle with a center at x = 3, y = 4 and a radius of 5.