Triangles- QA Formulae

Properties of a triangle:

1. A triangle has three sides, three angles, and three vertices.
2. The sum of all internal angles of a triangle is always equal to 180°. This is called the angle sum property of a triangle.
3. The sum of the length of any two sides of a triangle is greater than the length of the third side.
4. The side opposite the largest angle of a triangle is the largest side.
5. Any exterior angle of the triangle is equal to the sum of its interior opposite angles. This is called the exterior angle property of a triangle.

Based on the angle measurement, there are three types of triangles:

1. Acute angle triangle:

An acute angle triangle is a triangle in which all three angles are less than 90°.

2. Right angle triangle:

A triangle that has one angle that measures exactly 90° is a right-angle triangle.

• The other two angles of a right-angle triangle are acute angles.
• The side opposite the right angle is the largest side of the triangle and is called the hypotenuse.
  
  
  

  Pythagoras theorem:

  In a right-angled triangle, the sum of the squares of the perpendicular sides is equal to the square of the hypotenuse.

  For example, considering the right-angled triangle ACB, we can say:

  (AC)2 + (CB)2 = (AB)2

  Vice versa, we can say that if a triangle satisfies the Pythagoras condition, then it is a right-angled triangle.

  3. Obtuse angle triangle:

• A triangle that has one angle that measures more than 90° is an obtuse angle triangle.

Based on the length of the sides, triangles are classified into three types:

1. Scalene triangle:

A scalene triangle is a triangle in which all three sides are of different lengths.

• Since all the three sides are of different lengths, the three angles will also be different.

2. Isosceles triangle:

A triangle that has two sides of the same length and the third side of a different length is an isosceles triangle.

• The angles opposite to the equal sides measure the same.

3.Equilateral triangle:

A triangle which has all the three sides of the same length is an equilateral triangle.

• Since all the three sides are of the same length, all the three angles will also be equal.
• Each interior angle of an equilateral triangle is equal to 60°.

Median:

The line joining the midpoint of a side with the opposite vertex is called a median.

Altitude:

The perpendicular drawn from a vertex to the opposite side is called the altitude.

Perpendicular bisector:

A line that bisects and also makes a right angle with the same side of the triangle is called a perpendicular bisector.

Angle bisector:

The line that divides the angle at one of the vertices into two parts is called an angular bisector.

Note:

• All points on the angular bisector are equidistant from both arms of the angle.
• All points on the perpendicular bisector of a line are equidistant from both ends of the line.
• In an equilateral triangle, the perpendicular bisector, median, angle bisector, and altitude (drawn from a vertex to a side) coincide.

 Geometric centres

• Orthocentre:

The point of intersection of the three altitudes is the orthocentre.

• Centroid:

 The point of intersection of the three medians is the centroid.

• Circumcentre:

The three perpendicular bisectors of a triangle meet at a point called the circumcentre. A circle drawn from this point with the circumradius would pass through all the vertices of the triangle.

• Incentre: 

The three angle bisectors of a triangle meet at a point called the incentre of a triangle. The incentre is equidistant from the three sides, and a circle drawn from this point with the inradius would touch all the sides of the triangle.

Apollonius theorem

In a triangle ABC, if AD is the median to side BC then by Apollonius theorem, 

Midpoint theorem

The line joining the midpoint of any two sides in a triangle is parallel to the third side and is half the length of the third side. If X is the midpoint of CA and Y is the midpoint of CB, then XY will be parallel to AB and XY = ½ * AB.

Basic proportionality theorem

If a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points, then it divides the two sides in the ratio of respective sides.

If in a triangle ABC, D and E are the points lying on AB and BC, respectively, and DE is parallel to AC, then AD/DB = EC/BE.

Interior angular bisector theorem

In a triangle, the angular bisector of an angle divides the side opposite to the angle, in the ratio of the remaining two sides. In a triangle ABC, if AD is the angle bisector of angle A then AD divides the side BC in the same ratio as the other two sides of the triangle, i.e., BD/ CD= AB/AC. 

Exterior angular bisector theorem

The angular bisector of the exterior angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle. In a triangle ABC, if CE is the angular bisector of the exterior angle BCD, then AE/BE = AC/BC.

Similar triangles

 If two triangles are similar, then their corresponding angles are equal, and the corresponding sides will be in proportion. 

For any two similar triangles: 

• Ratio of sides = Ratio of medians = Ratio of heights = Ratio of circumradii = Ratio of angular bisectors 
• Ratio of areas = Ratio of the square of the sides

Tests of similarity: (AA/SSS / SAS)

Congruent triangles

If two triangles are congruent, then their corresponding angles and corresponding sides are equal. 

Tests of congruence: (SSS/SAS/AAS/ASA)

Area of a triangle, A

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