49 terms

johnsonjack580PLUS

Properties and Attributes of Triangles

'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)

(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)

(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)

(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)

(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)

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)

(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

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)

(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

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

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

The centroid is always inside the triangle

Centroid Theorem

(5-3-1)

(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

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

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

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)

(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

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

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)

(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)

(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)

(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

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

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)

(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

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)

(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)

(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