49 terms

Chapter 5 Holt McDougal Geometry Vocabulary

Properties and Attributes of Triangles

Terms in this set (...)

'equi' - root
equal
'co' - root
together with
'currere' - root
to run
'loci' - root
location (pronounced 'low-sigh' it is the plural of the word locus)
equidistant
When a point is the same distance apart at every point
locus
A set of points that satisfies a given condition
midpoint
The point that divides a segment into two congruent segments (the half way point)
point
An undefined term in Geometry, is simply names a location and has no size.
endpoint
A point at an end of a segment, or the starting point of a ray
vertex of a triangle
The intersection of two sides (rays) of a triangle
vertices of a triangle
Plural for Vertex the word refers to the 3 corners of a triangle
Perpendicular bisector of a segment
A line perpendicular to a segment located at the segments midpoint and forming a 90° angle
Perpendicular Bisector Theorem
(5-1-1)
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints on the segment
Converse of the Perpendicular Bisector Theorem
(5-1-2)
If a point is equidistant from the endpoints of a segment then it is on the perpendicular bisector of the segment
Angle Bisector Theorem
(5-1-3)
If a point is on the bisector of an angle, then it is equidistant from the sides (rays) of the angle
Converse Angle Bisector Theorem
(5-1-4)
If a point in the interior of an angle is equidistant from the sides (rays) of the angle, then it is on the bisector of the angle
Coincide
To correspond exactly; to be identical
Concurrent
Three or more lines that intersect at one point
Point of Concurrency
A point where three or more lines coincide; meet exactly at one point
circle
A set of points in a plane that are a fixed distance from a given point called the center of the circle
Circumscribed Circle
A circle that encloses a triangle, or other polygon and every vertex lies on the circle (touches the circle)
Circumcenter of a triangle
The point of concurrency (intersection) of the three perpendicular bisectors of a triangle.

It is the center of a circle circumscribed around the triangle

Can be constructed by finding the intersection of any 2 perpendicular bisections of a triangle (bisecting the midpoints of the sides with lines at 90° angles), which produces equidistant distances from the circumcenter to each vertices

The circumcenter can be inside a triangle (Acute), outside a triangle (Obtuse), or on a triangle (Right)
Circumcenter Theorem
(5-2-1)
The circumcenter of a triangle is equidistant from the vertices of the triangle
tangent
Intersecting at exactly one point
Inscribed circle
A circle placed inside a triangle, or other polygon touching each side at exactly one point (triangle sides are tangent to the circle)
Incenter of a triangle
The point of concurrency (intersection) of the three angle bisectors (vertices) of a triangle.

It is the center of a circle inscribed inside the triangle.

Can be constructed by finding the intersection of any 2 angle bisections of a triangle (bisecting the vertices to split the angles in half), which produces equidistant distances from the incenter to each side at their perpendicular midpoints

The incenter is always located inside the triangle
Incenter Theorem
(5-2-2)
The incenter of a triangle is equidistant from the sides of the triangle
Median of a triangle
A segment with one endpoint at a vertex and the other endpoint at the midpoint of an opposite side

Every triangle has 3 Medians
Centroid of a triangle
The point of concurrency (intersection) of the 3 Medians of a triangle.

Also known as the center of gravity (balancing point) of a triangle

The centroid is always inside the triangle
Centroid Theorem
(5-3-1)
The centroid of a triangle is located 2/3 of the distance of a Median (vertex to midpoint of opposite side). Therefore 1/3 of the Median distance remains to the opposite side
Altitude of a triangle
A perpendicular segment from a vertex to the line containing the opposite side.

Every triangle has 3 Altitudes

The height of a triangle is the length of an altitude

An altitude can be inside, outside, or on the triangle
Orthocenter of a triangle
The point of concurrency (intersection) of the 3 altitudes of a triangle

The orthocenter of a triangle can be inside, outside, or on the triangle
Midsegment of a triangle
A segment that joins the midpoints of two sides of the triangle

Every triangle has 3 Midsegments, which form a Midsegment triangle
Midsegment Triangle
A triangle formed by the three mid segments of a triangle
Triangle Midsegment Theorem
(5-4-1)
The midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side

Conversely, and easier for solving triangle problems, the side of a triangle parallel to its corresponding midsegment is twice the length of the midsegment
Direct Reasoning
The process of reasoning that begins with a true hypothesis and builds a logical argument to show that a conclusion is true
Indirect proof
A proof in which the statement to be proved is assumed to be false and a contradiction is shown.

Also called a Proof of Contradiction
Triangle angle-side relationships - The larger angle is opposite the longer side (5-5-1)
If two sides of a triangle are not congruent, then the larger angle is opposite the longer side
Triangle angle-side relationship - The longer side is opposite the larger angle (5-5-2)
If two angles of a triangle are not congruent, then the longer side is opposite the larger angle
Triangle angle-side relationship - The shorter side is opposite the smaller angle
If two angles of a triangle are not congruent, then shorter side lies opposite the smaller angle
Triangle Inequality Theorem
(5-5-3)
The sum of any two side lengths of a triangle is greater than the third side length
Inequalities in two triangles - Hinge Theorem
(5-6-1)
If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is across from the larger included angle.
Inequalities in two triangles - Converse Hinge Theorem
(5-6-2)
If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side.
Pythagorean triple
For a right triangle adding the square of the legs will equal the square of the hypotenuse
a² + b² = c²
Pythagorean Theorem
In a right triangle with leg lengths a & b and hypotenuse c, the sum of the squares of the lengths of legs (a & b) equals the square of the length of the hypotenuse (c)
c² = a² + b²
Converse of the Pythagorean Theorem (5-7-1)
If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle
Pythagorean Inequalities Theorem
(5-7-2)
In triangle where c is the length of the longest side.

If c² > a²+b², then the triangle is Obtuse

If c² < a²+b², then the triangle Acute
45°-45°-90° Triangle Theorem
(5-8-1)
In a 45°-45°-90° triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times √2.
30°-60°-90° Triangle Theorem
(5-8-2)
In a 30°-60°-90° triangle, the length of the hypothenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter length times the √3