Triangle

Definition

A triangle is a closed, three-sided polygon in a plane. It is one of the fundamental shapes in geometry and is often used to solve various problems related to spatial relationships, measurements, and applications.

5 Must Know Facts For Your Next Test

1. The sum of the interior angles of a triangle is always 180 degrees.
2. Triangles can be classified based on the lengths of their sides (equilateral, isosceles, scalene) or the measures of their angles (acute, obtuse, right).
3. The Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (longest side) is equal to the sum of the squares of the lengths of the other two sides, is a fundamental property of triangles.
4. Triangles are often used in the construction of stable structures, such as bridges and buildings, due to their inherent strength and rigidity.
5. The area of a triangle can be calculated using the formula: $A = \frac{1}{2} \times b \times h$, where $b$ is the base and $h$ is the height.

Review Questions

• Explain how the Pythagorean Theorem is used to solve problems involving right triangles.
  • The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. This relationship can be expressed as $a^2 + b^2 = c^2$, where $a$ and $b$ are the lengths of the shorter sides, and $c$ is the length of the hypotenuse. By using this theorem, we can solve for the unknown side length of a right triangle if the other two sides are known, or determine whether a triangle is a right triangle based on the lengths of its sides.
• Describe the different ways triangles can be classified based on their side lengths and angle measures.
  • Triangles can be classified based on the lengths of their sides or the measures of their angles. Based on side lengths, triangles can be classified as equilateral (all sides are equal), isosceles (two sides are equal), or scalene (no sides are equal). Based on angle measures, triangles can be classified as acute (all angles are less than 90 degrees), obtuse (one angle is greater than 90 degrees), or right (one angle is exactly 90 degrees). These classifications are important in solving various geometry problems and understanding the properties of different types of triangles.
• Explain how the area of a triangle can be calculated and how this formula is used to solve real-world problems.
  • The formula for calculating the area of a triangle is $A = \frac{1}{2} \times b \times h$, where $b$ is the base and $h$ is the height. This formula can be used to find the area of any triangle, regardless of its classification or the lengths of its sides. In real-world applications, the area formula for triangles is used to determine the space occupied by triangular structures, such as the roof of a building or the sail of a boat. It is also used in surveying, construction, and other fields where the measurement of triangular regions is necessary.

Related terms

Vertex:

The point where two or more line segments meet to form a corner of a polygon, such as a triangle.

Side: One of the three line segments that form the boundaries of a triangle.

Angle: The space between two intersecting line segments, measured in degrees or radians.