Triangles are polygons with three sides. It should be remembered that polygons are flat figures, delimited by segments (that is, by their sides). The triangle, therefore, is a plane figure composed of three segments.
When a triangle has a right angle (measuring ninety degrees), it is classified as a right triangle. The other two angles of the right triangle are always acute (less than 90 degrees). The right angle in the right triangle is formed by the two shorter sides, known as the legs, while the third (longer) side is called the hypotenuse. The properties of these triangles indicate that the length of the hypotenuse is always less than the sum of the legs. The hypotenuse, on the other hand, is always longer than any leg.
The famous Pythagorean theorem is based on these characteristics of right triangles and indicates that the square of the hypotenuse is identical to the result of the sum of the squares of the two legs. In this way, the following equation is established for each right triangle: Hypotenuse squared = Leg squared + Leg squared It should be noted that right triangles can be isosceles triangles (the two legs have the same length: that is, they are equal) or scalene triangles (the length of each side is different from the other two). On the other hand, if we want to calculate the area of a right triangle, we can use the following formula: Area = (Cat x Leg) / 2 As you can see, one of the fundamental points of triangles is the relationships that we can establish between their different sides and angles, something essential to solve a large number of problems, both in the field of mathematics and in many others. Before continuing with these relations, it is necessary to cover another topic: the orthogonal projection. The orthogonal projection belongs to the field of Euclidean geometry, which studies the geometric properties of the spaces in which Euclid’s axioms are fulfilled, a set of propositions considered evident that can generate others by logical deductions. To carry out an orthogonal projection, two elements are needed: a set of points (which can be composed of only one); a projection line. The first is projected onto the line with the help of auxiliary lines perpendicular to it, so that the resulting dimensions are correct only in one case: when a segment parallel to the line is projected.
This concept is often used in game development to create a false sense of depth, since no matter how far away objects are from the camera, they will always have the same dimensions on screen. Now, if we project the legs onto the hypotenuse in this way, we’ll get a geometric mean called the altitude with respect to the hypotenuse, a segment that starts at the point where the two legs meet and cuts the hypotenuse perpendicularly. When we plot the height relative to the hypotenuse, the right triangle becomes three triangles: the original plus the two it contains (as seen in the image). This results in certain metric relationships. For example, the sum of both projections is equal to the hypotenuse (a = m + n). It is also correct to say that the product of the two projections is equal to the square of the hypotenuse, since h/m = n/h, and solving for h gives us hh = mn. The product between the projection of a leg and the hypotenuse is equal to the square of said leg: b/a = m/b => bb = am. Finally, the product of the legs is equal to the relative height multiplied by the hypotenuse: a/c = b/h => ah = bc.
