Geometry Ch 5 - Relationships Within Triangles Flashcards
Midsegment of a Triangle
A segment connecting the midpoints of two sides.
Triangle Midsegment Theorem
If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length.
Midpoint Formula
Distance Formula
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
Distance from a Point to a Line
The length of the perpendicular segment from the point to the line.
Angle Bisector Theorem
If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.
Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.
Concurrent Lines
When three or more lines intersect in one point.
Point of Concurrency
The point at which concurrent lines intersect.
The perpendicular bisectors of the sides of a triangle…
…are congruent at a point equidistant from the vertices.
The bisectors of the angles of a triangle…
are concurrent at a point equidistant from the sides.
Circumcenter of the Triangle.
The point of concurrency of the perpendicular bisectors of a triangle.
Circumscribed (about)
To draw (a figure) around another, touching it at points but not cutting it.
Incenter of the Triangle
The point of concurrency of the angle bisectors of a triangle.
Inscribed (in)
To draw (a figure) within another so that their boundaries touch but do not intersect.
Median of a Triangle
A segment whose endpoints are a vertex and the midpoint of the opposite side.
The medians of a triangle are concurrent at…
… a point that is two thirds the distance from each vertex to the midpoint of the opposite side.
Centroid of a Triangle
In a triangle, the point of concurrency of the median. The point is also called the Center of Gravity of a triangle because it is the point where a triangular shape will balance.
Center of Balance of a Triangle
The centroid, because it is the point where a triangular shape will balance.
Altitude of a Triangle
The perpendicular segment from a vertex to the line containing the opposite side. Unlike angle bisectors and medians it can be a side of a triangle or it may lie outside the triangle.
Orthocenter of a Triangle
The point at which the lines containing the altitudes of a triangle are concurrent.
The lines that contain the altitudes of a triangle…
… are concurrent.
Negation of a Statement
Has the opposite truth value.
Inverse of a Conditional Statement
Negates both the hypothesis and the conclusion. A conditional and its inverse can (but may not) have different truth values.
Contrapositive of a Conditional Statement
Switches the hypothesis and the conclusion and negates both. It ALWAYS has the same truth value as the conditional. Thus, a conditional statement and its contrapositive are equivalent.
Equivalent Statements
Have the same truth value.
Comparison Property of Inequality
If a = b + c and c > 0, then a > b.
Corollary to the Triangle Exterior Angle Theorem
The measure of an exterior angle of a triangle is greater than the measure of the each of its remote interior angles.
If two sides of a triangle are not congruent…
… then the larger angle lies opposite the longer side.
If two angles of a triangle are not congruent…
…then the longer side lies opposite the larger angle. This is converse of Theorem 5-10.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
