Geometry Concepts: Theorems of Triangles

Hello and welcome to the third post of geometry concept series and in this article we will learn about the theorems of triangles. So let's start with the first one that is Pythagoras Theorem.

1. Pythagoras Theorem


According to this theorem, the square of the hypotenuse of a right angled triangle is equal to the sum of the sum of the squares of the other two sides


H² = B² + P²

AC² = AB² + BC². Its converse is also true.

The numbers which satisfy ths relation are called Pythagorean Triplets.

For eg. (5, 4, 3), (13, 12, 5), (25, 24, 7), (17, 15, 8), (41, 40, 9), (61, 60, 11), (37, 35, 12), (65, 63, 16), (29, 21, 20), (53, 45, 28), (65, 56, 33)

H² = B² + P²
  • (5)² = (4)² + (3)²
  • (13)² = (12)² + (5)²
  • (25)² = (24)² + (7)²
  • (17)² = (15)² + (8)²
  • (41)² = (40)² + (9)²
  • (61)² = (60)² + (11)²
  • (37)² = (35)² + (12)²
  • (65)² = (63)² + (16)²
  • (29)² = (21)² + (20)²
  • (53)² = (45)² + (28)²
  • (65)² = (56)² + (33)²

2. Triangle Theorem (45° - 45° - 90°)


If the angles of the triangle are 45°, 45° and 90°, then the hypotenuse is √2 times of any smaller side.

Excluding the hypotenuse rest of the two sides are equal.
∠A = 45°, ∠C = 45° and ∠B = 90°
That is....

AB =  BC
AC  = √2AB
AC  = √2BC

3. Triangle Theorem (30° - 60° - 90°)


If the angles of a triangle are 30°, 60° and 90° then the sides opposite to 30° angle is half of the hypotenuse and the side opposite to 60° is √3⁄2 times the hypotenuse e.g., AB = AC/2 and BC = √3⁄2AC
Therefore AB:BC:AC = 1:√3:2

4. Basic Proportionality Theorem (BPT) / Thales Theorem


Any line parallel to one side of a triangle divides the other two sides proportionally. So, if DE is drawn parallel to BC, it would divide sides AB and AC proportionally.
That is:

5. Mid-Point Theorem


If the mid-points of two adjacent sides of a triangle are joined by a line segment, then this segment is parallel to the third side, i.e., if AD = DB and AE = CE then DE | | BC.

6. Apollonius Theorem


In a triangle the sum of the squares of any two sides of a triangle is equal to twice the sum of the square of the of the median to the third side and square of half the third side i.e, AB² + AC²  =  2(AD² + BD²)
BD = CD and AD is the median

7. Interior Angle Bisector Theorem


In a triangle the angle bisector of an angle divides the opposite side to the angle in the ratio of the remaining two sides i.e., BD/CD = AB/BC and BD X AC - CD X AB = AD²

8. Exterior Angle Bisector Theorem


In a triangle the angle bisector of any exterior angle of a triangle divides the side opposite to the external angle in the ratio of the remaining two sides  i.e., BE/AE = BC/AC

No comments:

Post a Comment

Hello, This is Vikas (your mentor) and this is a quick tutorial as how to post your comment.

You can use your gmail account to post your comments and in case you do not have a gmail account. Then you have two options. You can either choose 'Name/URL' option or 'Anonymous' option from 'Comment as:' menu.

If you are using 'Name/URL' option:
Just fill your name in the 'Name' box and hit 'Continue'.

And after that just type in your comment in the box provided and hit 'Publish'.

If you are using 'Anonymous' option:
In this case you will need to type your name in the comment box and then type your comment and hit 'Publish' button.

Just remember comment moderation is ON that means your comments will be published after our approval. So that only useful comments and content is available on the website.

Thank you and good luck for your exams :)