Prisms, cylinders, pyramids, and cones is termed altitude. In addition, the height of the 3-dimensional objects:
The side opposite a vertex and contains that vertex.Īltitude can also refer to the length of the segment described above. If and only if it is perpendicular to the line containing The circumcenter as its center with a radius r, The circumcenter is equidistant (distance r) from all The three perpendicular bisectors of a triangle are coincident If and only if it contains the midpoint of the segment and is The incenter as its center with a radius r,Ī segment, ray, or line is a perpendicular bisector (of a segment) The incenter is equidistant (distance r) from all The three angle bisectors of a triangle are coincident at the incenter. If and only if it forms two angles of equal measure We will also discuss some specific applications of these lines to triangles. We will defined various important auxiliary lines Incenter, Circumcenter, Orthocenter, and Centroid In most triangles, these lines are different. Is an angle bisector, median, perpendicular bisector, and an altitude, If a triangle is equilateral, then it is equiangular.Ī corollary (a theorem which logically follows immediatelyĪngles of an equilateral triangle are all 60°.Īlthough the line of symmetry of an isosceles triangle These are the bisectors of the angles/sides. Historic Latin name: pons asinorum, or bridge ofĪsses/fools, due to the proof diagram that was used.Įvery equilateral triangle has three lines of symmetry. Then the sides opposite them are congruent) is also trueīut awaits chapter 7.
The converse (if two angles of a triangle are congruent, In an isosceles triangle, base angles are equal. Isosceles Triangle Base Angle Tneorem and commonly stated as: This last theorem is generally known as the If a triangle has two congruent sides, then the angles The perpendicular bisector of the base, and the median to In an isosceles triangle, the bisector of the vertex angle, The line containing the bisector of the vertex angle These can also be described as the angles at the endpoints of the base.Ĭertain terms will be defined further below. The two angles opposite the equal sides are the base angles (and are equal). The side opposite the vertex angle is called the base. The angle determined by the two equal sides There are special names associated with the angles and sides ofĪn isosceles triangle. Just as there are special names associated with the sides of Note: a right/acute/obtuse triangle might be either scalene or isosceles.Īlso, our definition of isosceles includes and does not exclude the Or by the largest angle (acute, right, obtuse).Ī hierarchy chart combining both situations is given at the left.ĭue to the overlap, hierarchy charts for either situation (0 is scalene, 2 or more is isosceles, all 3 is equilateral) Symmetric Triangles (Isosceles and Equilateral) The function y= x 2 defines a parabola in which These symmetries will be useful when applied to various polygons. Rorschach inkblots and logos commonly are reflective-symmetric. If a figure is symmetric, then any pair of corresponding parts The section concludes with the following important result. The semicircles are reflections of each other). (no matter which way you draw the diameter, This is the same as the letter I discussed above.Īngles only have one line of symmetry: the angleīisector which causes one ray to reflect onto the other ray.Ī circle has infinitely many lines of symmetry Our textbook states and proves what they call theįlip-Flop Theorem: (reflection is symmetric). Them into a crossword puzzle (for extra credit)! Of letters with a vertical line of symmetry.Īfter collecting enough of these words you might make Or MOM, WAXY, YOUTH (written vertically!) composed entirely There may be more than one line of symmetry.Ī challenge would be to find words such as DIXIE or COOKBOOKĬomposed entirely of letters with a horizontal line of symmetry Whereas others are symmetric about a vertical line (AHIMOTUVWXY).Īs you can see since some are in both lists (HIOX), Some are symmetric about a horizontal line (BCDEHIKOX) The capital letters A, B, C, D, E, H, I, K, M, O, T, U, V, W, X, and YĪre often written as reflection symmetric figures. This line is a symmetry line for the figure. If and only if there is a line which reflects the figure onto itself.
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