Triangle Types and Properties Quiz

Definition

• A triangle is a polygon with three edges and three vertices.

Types of Triangles

1. By Sides:

   • Equilateral Triangle: All sides are equal, and all angles are 60°.
   • Isosceles Triangle: Two sides are equal, and the angles opposite those sides are equal.
   • Scalene Triangle: All sides and angles are different.

2. By Angles:

   • Acute Triangle: All angles are less than 90°.
   • Right Triangle: One angle is exactly 90°; it has a hypotenuse and two legs.
   • Obtuse Triangle: One angle is greater than 90°.

Properties

• The sum of interior angles is always 180°.
• The exterior angle is equal to the sum of the two opposite interior angles.
• In a right triangle, the Pythagorean theorem applies: ( a^2 + b^2 = c^2 ) (where ( c ) is the hypotenuse).

Area Calculation

• Base and Height Method:

  • Area = ( \frac{1}{2} \times \text{base} \times \text{height} )

• Heron's Formula:

  • Area = ( \sqrt{s(s-a)(s-b)(s-c)} )
  • Where ( s = \frac{a+b+c}{2} )

Perimeter

• Perimeter = ( a + b + c ) (sum of all sides).

Special Points

• Centroid: Intersection of medians; balance point of the triangle.
• Circumcenter: Intersection of perpendicular bisectors; center of the circumcircle.
• Incenter: Intersection of angle bisectors; center of the incircle.
• Orthocenter: Intersection of altitudes.

Triangle Inequality Theorem

• The sum of the lengths of any two sides must be greater than the length of the third side.

Applications

• Used in various fields such as architecture, engineering, and computer graphics.
• Fundamental in trigonometry for defining sine, cosine, and tangent functions.

Definition

• A triangle is a polygon characterized by three edges and three vertices.

Types of Triangles

• By Sides:

  • Equilateral Triangle: Features three equal sides and angles of 60° each.
  • Isosceles Triangle: Has two equal sides and the angles opposite these sides are also equal.
  • Scalene Triangle: All sides and angles vary in length and measure with no equalities.

• By Angles:

  • Acute Triangle: All angles measure less than 90°.
  • Right Triangle: One angle measures exactly 90°; comprises a hypotenuse and two legs.
  • Obtuse Triangle: Contains one angle that exceeds 90°.

Properties

• The interior angles of any triangle sum to 180°.
• The exterior angle of a triangle equals the sum of its two opposite interior angles.
• The Pythagorean theorem applies to right triangles, expressed as ( a^2 + b^2 = c^2 ) where ( c ) designates the hypotenuse.

Area Calculation

• Base and Height Method:
  • Area = ( \frac{1}{2} \times \text{base} \times \text{height} )
• Heron's Formula:
  • Area = ( \sqrt{s(s-a)(s-b)(s-c)} )
  • Where ( s = \frac{a+b+c}{2} ), the semi-perimeter of the triangle.

Perimeter

• The perimeter of a triangle is calculated as the sum of its sides: ( a + b + c ).

Special Points

• Centroid: The point where all three medians intersect, acting as the triangle's balance point.
• Circumcenter: The intersection point of the perpendicular bisectors, which serves as the center of the circumcircle.
• Incenter: The intersection of the angle bisectors, representing the center of the incircle.
• Orthocenter: The point where the altitudes of the triangle intersect.

Triangle Inequality Theorem

• States that the sum of the lengths of any two sides must surpass the length of the remaining side.

Applications

• Triangles are extensively utilized in architecture, engineering, and computer graphics.
• Fundamental in trigonometry for defining sine, cosine, and tangent functions.