Triangle Properties

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Triangle Properties by Mind Map: Triangle Properties

1. Triangle Side/Angle Relationships

1.1. Triangle Inequality Theorem

1.1.1. The sum of any two side lengths of a triangle is greater than the third side length.

1.2. Angle-Side Relationships

1.2.1. The largest angle of a triangle is opposite its longest side.

1.2.2. The smallest angle of a triangle is opposite its shortest side.

1.3. Hinge Theorem

1.3.1. Assuming that two sides of one triangle are congruent to two sides of another triangle:

1.3.1.1. The triangle with the longer third side has the larger included angle.

1.3.1.2. The triangle with the larger included angle has the longer third side.

2. Pythagorean Theorem

2.1. In a right triangle, a^2 + b^2 = c^2.

2.1.1. If the side lengths of a right triangle are integers, then they form a Pythagorean Triple.

2.2. Right Triangles

2.2.1. Special Right Triangles

2.2.1.1. 45°-45°-90°

2.2.1.1.1. When the legs are of length n, the hypotenuse is of length n√2.

2.2.1.2. 30°-60°-90°

2.2.1.2.1. When the shortest leg is of length n, the longest leg is of length n√3.

2.2.1.2.2. When the shortest leg is of length n, the hypotenuse is of length 2n.

2.2.2. If c^2 = a^2 + b^2, then △ABC is a right triangle.

2.3. Acute Triangles

2.3.1. If c^2 < a^2 + b^2, then △ABC is an acute triangle.

2.4. Obtuse Triangles

2.4.1. If c^2 > a^2 + b^2, then △ABC is an obtuse triangle.

3. Lines in Triangles

3.1. Median

3.1.1. A median is a line joining a vertex and the midpoint of its opposing side.

3.2. Altitude

3.2.1. An altitude is a line through a vertex that is perpendicular to its opposing side.

3.3. Perpendicular Bisector

3.3.1. A perpendicular bisector is a line passing through the midpoint of a side, perpendicular to that side.

3.3.2. Distance

3.3.2.1. If a point is on the perpendicular bisector of a side, then it is equidistant from the side's endpoints.

3.3.2.2. If a point is equidistant from a side's endpoints, then it is on the perpendicular bisector of the side.

3.4. Angle Bisector

3.4.1. An angle bisector is a line passing through a vertex that passes directly in between the angle's adjacent sides.

3.5. Midsegment

3.5.1. A line that is parallel to a side of the triangle, with a length that is half the length of that side.

4. Points of Concurrency (Loci)

4.1. Centroid

4.1.1. The point where the triangle's medians meet is called the Centroid.

4.1.2. The Centroid of a triangle is two-thirds of the distance from each vertex to the midpoint of the opposite side.

4.1.3. The Centroid of a triangle is its center of balance.

4.2. Orthocenter

4.2.1. The point where the triangle's altitudes meet is called the Orthocenter.

4.3. Circumcenter

4.3.1. The point where the triangle's perpendicular bisectors meet is called the Circumcenter.

4.3.2. Circumcircle

4.3.2.1. The Circumcircle is outside of the triangle and touches all of the vertices.

4.3.2.2. It is centered on the Circumcenter.

4.3.2.3. This shows that the distances from the Circumcenter to each of the vertices are equal, as the radius of the Circumcircle is the same at all points.

4.4. Incenter

4.4.1. The point where the triangle's angle bisectors meet is called the Incenter.

4.4.2. Incircle

4.4.2.1. The Incircle is inside of the triangle and touches all of the sides once.

4.4.2.2. It is centered on the Incenter.

4.4.2.3. This shows that the shortest distances from the Incenter to each of the sides are equal, as the radius of the Incircle is the same at all points.

4.5. Geometry - GeoGebra

5. In an Indirect Proof, you must prove that the negation of the conclusion leads to a contradiction.