math [1.06-01] the philosophy of triangles

three triangles, two interior, in bright green and titled for the philosophy of triangles page at hiartx.com


A triangle is a shape that consists of three distinct points, with three distinct angular verticies, joined by three distinct lines.

The large green triangle to the right is a right triangle, meaning that it has one vertex that is exactly 90 degrees.

The two lines that make the right angle are perpendicular.

The area of the triangle is equivalent to 1/2 the length of the base of the triangle multiplied by the height of the triangle. [ A = 1/2 bh ].

The dark green shaded rectangle in the image to the right represents the area of the triangle formula visualized.

Notice that the shaded area that is not part of the triangle, to the upper right of the rectangle, is the same area as the part of the triangle that is not shaded.

Angle A, written [ A], is equivalent to angle B. We can write this as [ A = B ].

The squares in each triangle's corner represent that the corner is a 90 degree angle, or right angle, and that the triangle is a right triangle.

Right triangles and the relationship between their sides and angles are the foundations of trigonometry and trigonometric functions.

In the image to the right, there is one large green triangle and four smaller triangles for a total of five triangles.

 
large triangle graphic showing the area of the triangle as compared to the area of a rectangle that is one half the base times the height of the triangle

It is hard to imagine that any triangle could exist in reality in that the points and lines must be inserted between places and there are no perfectly straight lines in reality. Also, points for the vertices are only theoretical as they exist in the mathematical mental world but not in the real world. For more on points go to the points page linked here.


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