PROPERTIES AND ATTRIBUTES OF TRIANGLES

By: Mariana Beltranena 9-5PROPERTIES AND ATTRIBUTES OF TRIANGLES

Perpendicular BisectorThe perpendicular bisector of a segment is aligned perpendicular to other segment and to its midpoint.

Perpendicular Bisector TheoremIf a point is in the perpendicular bisector of a segment, then it is equidistant from both endpoints.

and its Converse If a point is equidistant from both endpoints of a segment then it is the perpendicular bisector of a segment.

Angle BisectorIs a line that divides an angle into two congruent angles.

Angle Bisector TheoremIf a point is on the angle bisector of an angle then the perpendicular distance to each side of the angle is the same.

and its ConverseIf the perpendicular distance from a point to both sides of an angle is the same, then the point is on the angle bisector of that angle.

Concurrency of Perpendicular BisectorsConcurrent means the coincidence on a point. Where three or more lines intersect at one point.

The concurrency of bisectors is when the three perpendicular bisectors of a triangle are on a point called circumcenter. This point can be inside or outside of the triangle. Also around the triangle it could be drawn a circle touching all of the corners.

Circumcenter TheroremThe circumcenter is equidistant from the 3 vertices of the triangle.

Concurrency of Angle BisectorsThe three angle bisectors of a triangle are concurrent on a point called incenter, which is always in the triangle.

Incenter Theorem Is always in and is where the angle bisector meets. The perpendicular distance from the incenter to the three sides of the triangle is the same.

MediansThe median of a side of a triangle is the line from the midpoint of that side to the vertex opposite to it.Centroid: the point of concurrency of the medians of a triangle. The centroid is always inside of the triangle.

Centroid TheoremThe distance from the vertex to the centroid is 2/3 of the distance from the vertex to the opposite side midpoint.

Concurrency of altitudes of a triangle theorem The three altitudes of a triangle are concurrent in a point called orthocenter which has no special properties.The altitude of a triangle is the line from the vertex to its opposite side or the prolongation of that side if the triangle is obtuse.

MidsegmentsThe midsegment of a triangle is the line that joins the midpoints of two of its sides.Midsegment TheoremThe midsegment of a triangle is parallel to the other side and measures half the measure of that side. Midsegment examples

the relationship between the longer and shorter sides of a triangle and their opposite angles In the same triangle or in congruent triangles with no congruent angles the side opposite to the biggest angle is the biggest and the side opposite to the smallest angle is the smallest side.

Triangle InequalityIn a triangle inequality the sum of two sides is always greater than the third side length.

Triangle inequality examplesCan 8,6,10 be the measures of a triangle? If so tell if it is acute obtuse or right.8+6>1014>10Yes8+6=1064+36=100100=100It is a right triangleCan 5,6,11 be a triangle?5+6 11.11= 11. No, because the two short sides have to add up a greater number than the longer side. The measures of the sides of a triangle are 5 and 9. Find all the possible measures of the third side.5+9= 149-5=3x>3, x