Depending on the age of your students and the time alloted for this lesson, you may want. Thus, in hyperbolic geometry, congruence is the same as similarity, eliminating the need for similarity entirely. 2 paper squares of different colors for the Pythagorean Theorem proof. In fact the only way two triangles could have their angles the same and their sides in a ratio is if they are the same triangle altogether (the ratio would be 1:1). Even if the side lengths of two triangles are in a ratio, the angles of the larger triangle will be smaller, thus making them not similar. (The area can also be some multiple of this, depending on what units of measurement you're using.)Īs a corollary of this, because one would expect the area to get larger as you increase the side lengths, this theorem also implies that the angles in a triangle get smaller as the side lengths get longer.īuilding off that last statement, you can now say that there is no such thing as similar triangles in hyperbolic geometry. The Spherical Pythagorean Theorem is derived by performing an infinitesimal rotation of a spherical right triangle about a vertex not containing the right.
Thus the maximum area of a triangle approaches 180° = π. Thus, the maximum area is when all the angles are as close to zero as possible, thus making the the defect approximately 180°. The defect of a triangle ABC - written as δ(ΔABC) and calculated as 180°-(sum of the angles) - is equal to its area in hyperbolic geometry. In hyperbolic geometry, the sum of the angles in a triangle is always less than 180°.
In hyperbolic geometry, the area of a triangle is given by its defect.
0 kommentar(er)
