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Larson Geometry Reference – Chapter 6 Definitions Term/Concept *Altitude Centroid Circumcenter Description A line segment from a vertex of a triangle to the side opposite the vertex, perpendicular to that side. The point of concurrency of the medians of a triangle. The point of concurrency of three perpendicular bisectors of a triangle. B G is the circumcenter of ABC. Therefore AG BG CG because they are radii Of the circle going through the vertices A, B, and C. Concurrent *Equidistant Incenter Inequality *Median of a Triangle *Midsegment of a Triangle Orthocenter of a Triangle *Perpendicular Bisector *Point of Concurrency E D G A F C When three or more lines, rays, or segments intersecting at the same point they are called concurrent. That point is the point of concurrency. The property of two or more objects being the same distance from an another object. The point of concurrency of the angle bisectors of a triangle. For any real numbers a and b, a > b if and only if there is a positive number c such that a = b + c. A line segment that connects a vertex of a triangle to the midpoint of the side opposite that vertex. The segment that connects the midpoints of two sides of a triangle. The point at which the lines containing the three altitudes of a triangle intersect. The point of concurrency of the three altitudes of a triangle. A segment, ray, line, or plane that is perpendicular to a segment at its midpoint. The point at which concurrent lines rays or segments intersect. Larson Geometry Reference – Chapter 6 Theorems Theorems *Angle Bisector Theorem *Centroid Theorem Circumcenter Theorem Concurrency of Altitudes of a Triangle Concurrency of Angle Bisectors of a Triangle Concurrency of Medians of a Triangle Concurrency of Perpendicular Bisectors of a Triangle Converse of the Angle Bisector Theorem Converse of the Perpendicular Bisector Theorem Converse of the Hinge Theorem Hinge Theorem *Incenter Theorem Orthocenter Theorem *Perpendicular Bisector Theorem Triangle Longer Side Theorem Triangle Larger Angle Theorem *Triangle Inequality Theorem *Triangle Midsegment Theorem Any point on the bisector of an angle is equidistant from the sides of the angle. The centroid of a triangle is two-thirds the distance from the vertex to the midpoint of the opposite side along a median. The circumcenter of a triangle is equidistant from the vertices of the triangle. The lines containing the altitudes of a triangle are concurrent. The angle bisectors of a triangle intersect at ta point that is equidistant from the sides of the triangle. The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. Any point on or in the interior of an angle and equidistant from the sides of an angle lies on the bisector of the angle. In a plane, any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment. If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second. If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second. The incenter of a triangle is equidistant from the sides of the triangle. The lines containing the altitudes of a triangle are concurrent. In a plane, any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side.