Interspersing Geometry With Algebra

Let us study some properties derived in Geometry using algebra.

Let us take the example of a straight line. What do we observe? A straight line intersects the X-Axis or the Y-Axis in one of the four quadrants. A line can be plotted hanging somewhere in the middle, but dragging it either way would make it certainly intersect in one of the four quadrants. What are the properties of a straight line? A straight line intersects either the x-axis or the y-axis with an angle. If this line makes an angle of 90 degrees with the X-Axis, then it is parallel to the Y axis or the Y-Axis itself. On the contrary if this line makes an angle of 90 degrees with the Y-Axis then it runs parallel to the X-Axis or can be the X-Axis itself.

Let us take a point on the line as (X,Y), let us investigate the relationship between X and Y. Let us project the point to the X and the Y axis respectively. Let the line intersect on the X-Axis at some point (C1,0) and the Y-Axis at point (0,C).

Let us consider the right triangle between the origin and the two intersection points on the X and the Y axis(where the straight line meets the two axis. Let theta be the angle made by the straight line and the X-Axis. By definition tan(theta) is equal to height /base of a right triangle. So tan(theta) in this case is nothing but C/C1.

At any other point (X,Y) on the straight line tan(theta) is equal to Y/C1-X.

Equating both we get Y/C1-X= C/C1 so Y = C(C1-X)/C1 = -XC/C1 + C.

Since theta is the interior angle made by the straight line with the X-Axis, the exterior angle is equal to PI-Theta. Also, tan(theta) = -tan(PI-theta).

So if follows that -C/C1 = tan(exterior angle).

Y = tan(exterior angle) * X + C. This is just the popular equation Y = M*X + C.

Now let us apply some elementary algebra to derive the pythogoreas’ theorem.

Let us consider a right triangle at the origin with coordinates (0,0), (a,0) and(0,b)

The length of the hypotenuse is nothing but sqrt (a*a + b*b ).

This is just the sum of the squares of the other two sides, which is as per the Pythogoreas’ theorem.

Now let us move to a circle, what are the properties of a circle. Any point along the circle is at a distance of r from the center of the circle. Let the center of the circle be at the origin. Let us take a point (X,Y) located at any point on a circle. So the distance of that point to the center is nothing sqrt(X *X + Y * Y) which is equal to r the length of the radius.

So the equation of a circle is sqrt(X*X + Y*Y) = r or X*X + Y*Y = r*r.

Applying Algebra to Geometry is popularly termed as co-ordinate geometry.

Source by Srinivasa Gopal

Leave a Reply