The following text can be an example of the homework given in a CPM program. It reveals the approaches on the narration of theoretical and practical parts of the assignment.
The analytical geometry gives the same approaches to problem-solving. For this, all defined and searched points and lines are referred to as one frame of reference.
In this frame, each point can be characterized by its coordinates and each line – by the equation with two unknowns, whose graphic forms this line.
That is how a geometrical problem is turned into arithmetical where all calculation approaches are set well. A circle is a geometrical place of points with one defining feature (each point of the circle is situated on the same distance from the point called the center). The equation of the circle must reflect this feature and fulfill this condition.
The geometrical interpretation of the circle equation is the line of a circle. If a circle is placed into the frame of reference all points of the circle run one option – the distance between them and the center of the circle must be equal.
The circle with the center in point A and radius R is placed into the coordinate frame. If the coordinates of the center are (a;b) and the coordinates of every point of the circle are (x;y) then the equation of the circle looks like (x-a)2+(y-b)2=R2
If the square of the circle radius is equal to the sum of the square diminutions of correlated coordinates of each point of the circle and its center this equation appears to be the equation for a circle in the plain coordinate system.
If the center of the circle lays on the point of the beginning of coordinates, the square of the radius of the circle equals the sum of square coordinates of each point of the circle. In this case, the circle equation looks like x2+y2=R2
So every geometric shape is defined by the equation which ties the coordinates of its points. And vice versa, the equation which connects coordinates x and y defines the line as the geometrical place of the points of the plane, the coordinates of which are equal to the given equation.
Let’s check the following problem:
Requirement: Define the radius of the circle if it is 107 cm longer than its diameter.
Solution: Mark the length of the circle as C and diameter as D. So C-D=107sm
The length of the circle equals С = 2πR = πD, so
πD – D = 107
D ( π – 1 ) = 107
D = 107 / ( π – 1 ) ≈ 49,96 sm
The radius of the circle is R = D / 2 = 107 / 2( π – 1 ) ≈ 24,98 sm
Answer: 107 / 2( π – 1 ) ≈ 24,98 sm
- Johnson, Roger A. Advanced Euclidean Geometry. Dover Publications, 1960. Scribd, https://ru.scribd.com/document/328407159/Advanced-Euclidean-Geometry-Roger-Johnson-Dover-1960-pdf. Accessed 23 Apr. 2018.
- Cook, Theodor A. The Curves of Life. Dover Publications, 1914. Avalonlibrary, http://avalonlibrary.net/ebooks/Theodore%20Cook%20-%20The%20Curves%20of%20Life.pdf. Accessed 23 Apr. 2018.
- “Basic Equation of a Circle.” Math Open Reference, https://www.mathopenref.com/coordbasiccircle.html. Accessed 23 Apr. 2018.