Exploring Relationships Within Triangles Class Lectures

The STEM concept of relationships within triangles geometry focuses on the various relationships that exist within triangles. These relationships include the Pythagorean theorem, which states that the sum of the squares of the lengths of the two shorter sides of a right triangle is equal to the square of the length of the hypotenuse. Another relationship is the Law of Sines, which relates the lengths of the sides of any triangle to the sines of its angles. The Law of Cosines is also a relationship within triangles geometry, which relates the lengths of the sides of a triangle to the cosine of one of its angles. Understanding these relationships is essential for solving problems in trigonometry and geometry. By applying these relationships, mathematicians and scientists can calculate the measurements of triangles and use them to solve real-world problems.

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Exploring Relationships Within Triangles

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Exploring Relationships Within Triangles Lecture Videos, Solved Step-by-Step

Exploring Relationships Within Triangles

Altitude Example

Altitude or height is defined based on the context in which it is used (as a height above sea level, or as height above the ground, or as height above the ground or sea level). As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context. Although the term altitude is commonly used to mean the height above sea level of a location, in geography the term "elevation" is often preferred for this usage.

Kurt Kleinberg

Exploring Relationships Within Triangles

Altitudes

In mathematics, an altitude is the perpendicular distance from a geometric figure to its line of intersection with a horizontal plane. The word altitude comes from the Latin word "altitudo", meaning "height".

Kurt Kleinberg

Exploring Relationships Within Triangles

Angle Bisector Examples

In mathematics, an angle bisector of a polygon is a line segment dividing the angle of the polygon in two equal angles. It is a special case of the line bisector of a segment.

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Exploring Relationships Within Triangles

Angle Bisectors and Incenters

In geometry, the bisector of an angle of a triangle is a line segment that bisects the angle, that is, divides it into two equal angles. In a triangle, the bisector of an angle is the line through the vertex that divides the opposite side into segments with lengths proportional to the adjacent sides. In a circle, the bisector of an angle is a chord that passes through the center of the circle and divides the angle in half. The incenter of a triangle is the center of its incircle, the circle passing through all three vertices of the triangle.

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Exploring Relationships Within Triangles

Identifying Congruent Triangles Quick

In geometry, two triangles are congruent if they have the same shape and size. The word is occasionally used to refer to all similar triangles. Two triangles are similar if both pairs of corresponding sides are in the same ratio, and the included angles also match.

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Exploring Relationships Within Triangles

Intro to Congruent Triangles

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted ? ABC. In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane (the triangle's plane). This article is about triangles in Euclidean geometry, and in other geometries.

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Exploring Relationships Within Triangles

Isosceles and Equilateral Triangles

An isosceles triangle is a triangle in which at least two sides have the same length. An equilateral triangle is a triangle in which all three sides have the same length.

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Exploring Relationships Within Triangles

Isoscelese and Equilateral Examples

In geometry, an isosceles triangle is a triangle in which two sides have equal lengths. The term isosceles triangle is also used to refer to any triangle that has at least two congruent sides, regardless of the lengths of the other two sides. The base angles of an isosceles triangle are also equal. An equilateral triangle is a special case of an isosceles triangle with all three sides equal in length, and equiangular triangle is a special case of isosceles triangle with all three angles equal.

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Exploring Relationships Within Triangles

Medians Centroid and Examples

In mathematics, the median is a measure of the "middle" value of a data set. The median is a commonly used measure of the properties of a set in statistics and probability theory. The median of a finite list of numbers can be found by arranging all the numbers from smallest to greatest, and finding the middle one. For example, the median of the list [3, 4, 6, 7, 8, 9] is 6. If there is an even number of elements, then there is no single middle value. In this case, one may take the mean of the two middle values. For example, the median of the list [3, 4, 4, 6, 7, 8, 9] is (4+6)/2 = 4.5.

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Exploring Relationships Within Triangles

Proving Triangles Congruent Cont

In mathematics, two triangles are congruent if they have the same shape and size. More formally, two triangles are congruent if both of the following statements are true: 1) the triangles have two corresponding angles, and 2) the corresponding sides have the same length. The first statement says that the triangles share the same angles. The second statement says that the triangles share the same sides. The corresponding sides are the pairs of sides that are connected to the same vertex. The corresponding angles are the pairs of angles that are opposite the same side.

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Exploring Relationships Within Triangles

Proving Triangles Congruent Proofs

In geometry, two triangles are congruent if they have the same size and shape, or if one can be transformed into the other by a combination of rotations, reflections, and translations.

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Exploring Relationships Within Triangles

Similar Triangle Examples with Algebra

In geometry, similarity is a relationship between two geometric figures. The precise definition depends on the type of figures being considered.

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Exploring Relationships Within Triangles

Similar Triangles

In geometry, similarity is a relationship between two geometric figures. The similarity transformation is a rigid transformation, i.e. a transformation that preserves the lengths of all vectors (or, in other words, the areas of all figures).

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Exploring Relationships Within Triangles

Similar Triangles Proofs

In geometry, two triangles are said to be similar if they have the same angles, and the corresponding sides are in proportion. Two triangles are similar if their corresponding angles are equal.

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Exploring Relationships Within Triangles

Slope Formalized

In mathematics, a slope or gradient of a line is a number that describes both the direction and the steepness of the line. The direction of a line is increasing, decreasing, horizontal, or vertical. The steepness is how much the line changes direction over a given distance. The slope can be expressed in several ways: as an angle made with the horizontal, as a ratio of distances, as a fraction, as a percentage, or as a vector.

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Exploring Relationships Within Triangles

Triangle Classifications

In mathematics, triangle classification is the classification of triangles into categories based on their properties. The triangle classification problem was first posed by Hilbert in 1902 and solved in the negative in 1983 by Robertson and Seymour. Classification of triangles is an important problem in computer vision and image processing, and is used for example in image analysis and automatic optical inspection.

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Exploring Relationships Within Triangles

Triangle Congruence Theorems

In geometry, two triangles are congruent if they have the same shape and size. More precisely, two triangles are congruent if one can be moved so that it exactly covers the other, with one side of the one matching up with the corresponding side of the other. The word congruent is derived from the Latin congruus, meaning "agreeing in length" or "of the same length".

Kurt Kleinberg

Exploring Relationships Within Triangles

Triangle Inequality Theorem Algebraic Example

In mathematics, the triangle inequality states that for any triangle "ABC", the length of any side must be less than the sum of the other two sides or the other two sides must be greater than the length of the given side. The triangle inequality is a basic property of Euclidean geometry, and is used to prove theorems such as the Pythagorean theorem and the law of cosines.

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Exploring Relationships Within Triangles

Triangle Inequality Theorem and Examples

In mathematics, the triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side, if the triangle is not isosceles. The inequality is a consequence of the fact that the two shorter sides of a triangle must be adjacent (that is, one side of the triangle must be between the other two sides).

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