Relationships Within Triangles Flashcards

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Questions and Answers

What is the concurrency of angle bisectors of a triangle?

The incenter of a triangle is equidistant from the sides of the triangle.

What is a median in the context of triangles?

A segment that connects a vertex to the midpoint of the opposite side.

What does the concurrency of altitudes of a triangle indicate?

The 3 altitudes of a triangle are concurrent.

What is an altitude in triangle geometry?

The perpendicular segment from a vertex of a triangle to the opposite side. Signup and view all the answers

What is a centroid?

A common point of the 3 medians, balancing point of the triangle, always located inside the triangle, separates medians into segments with a 2:1 ratio. Signup and view all the answers

What is a perpendicular bisector?

A segment, line, or ray that is perpendicular to a segment at its midpoint. Signup and view all the answers

What is a midsegment in a triangle?

A segment that connects the midpoints of 2 sides of a triangle. Signup and view all the answers

What is the converse of the angle bisector theorem?

If a point is in the interior of an angle and it is equidistant from the sides of the angle, then it lies on the bisector of the angle. Signup and view all the answers

What is a circumcenter?

The common point of the perpendicular bisectors, equidistant from the vertices of a triangle, center of the circumscribed circle, may be located outside or on the triangle. Signup and view all the answers

What does the midsegment theorem state?

The midsegment is parallel to the third side and half as long as that side. Signup and view all the answers

What is the converse of the perpendicular bisector theorem?

If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment. Signup and view all the answers

What does the perpendicular bisector theorem state?

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Signup and view all the answers

What is an angle bisector?

A line intersecting an angle into equal parts. Signup and view all the answers

What does the angle bisector theorem state?

If a point is on the angle bisector of an angle, then it is equidistant from the two sides of the angle. Signup and view all the answers

What is an orthocenter?

The common point of all three altitudes, may be on or outside the triangle (acute-inside, right-on the right angle, obtuse-outside). Signup and view all the answers

What does equidistant refer to in geometry?

The same distance from a figure or figures. Signup and view all the answers

What is concurrency in triangle geometry?

Three or more lines that intersect at a common point, referred to as the point of intersection. Signup and view all the answers

What does the concurrency of medians of a triangle indicate?

The centroid of a triangle separates the median into segments that are in the ratio 2:1. Signup and view all the answers

What is an incenter?

The common point of the angle bisectors, center of the inscribed circle, always located inside the triangle, equidistant from the sides of the triangle. Signup and view all the answers

What does the concurrency of perpendicular bisectors indicate?

The circumcenter is equidistant from the vertices of a triangle. Signup and view all the answers

Study Notes

Concurrency in Triangles

The incenter is equidistant from the triangle's sides, marking concurrency of angle bisectors.
Circumcenter serves as the common point for perpendicular bisectors, being equidistant from triangle vertices.
Centroid acts as the convergence point for all medians, creating a 2:1 ratio between segments.

Key Triangle Segments

A median connects a vertex to the midpoint of the opposite side, dividing the triangle.
An altitude is a perpendicular segment from a vertex to the opposite side.
A midsegment links midpoints of two triangle sides, maintaining parallelism to the third side.

Angle and Perpendicular Line Properties

The angle bisector divides an angle into two equal parts, providing necessary angles for further calculations.
The angle bisector theorem affirms points on the bisector are equidistant from both angle sides.
The perpendicular bisector is perpendicular at the midpoint of a segment, ensuring equal distance from segment endpoints.

Theorems and Definitions

The midsegment theorem illustrates that the midsegment is parallel to the third side and half its length.
The converse of the angle bisector theorem states that a point equidistant from angle sides lies on the bisector.
The converse of the perpendicular bisector theorem posits that a point equidistant from segment endpoints resides on the segment’s bisector.

Special Points in Triangles

The orthocenter is the concurrent point of altitudes; its position varies based on triangle type (acute, right, obtuse).
The incenter is central to the inscribed circle, always located within the triangle and equidistant from its sides.
Equidistant refers to being the same distance from one or more figures, crucial in defining concurrency and centers within triangles.

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Description

Explore key concepts related to the relationships within triangles through this flashcard set. Learn about angle bisectors, medians, altitudes, and their properties. Ideal for students seeking to deepen their understanding of triangle geometry.