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Triangle Part - II ( Advanced)

If two triangles are congruent they have equal sides, equal areas. 

Condition for Congruence
SAS Theorem:
If two sides and the angle between these two sides (included angle) of one triangle are congruent to the corresponding two sides and the included angle of a saecond triangle, then the two triangles are congruent

Example:



AAS Theorem:
If two angles and a side not between these two angles of one triangle are congruent to two angles and the corresponding side not between these two angles of a second triangle, then the two triangles are congruent

Example:


Hypotenuse-Leg Theorem:

If the hypotenuse and leg of a right triangle are congruent to the hypotenuse and corresponding leg of a second right triangle, then the two triangles are congruent

Example:



Pythagoras’ Theorem: In a right angle triangle the square of the hypotenuse is equal to the sum of the squares of other two sides.



b2=a2+c2

Pythagorean Triplet: There are certain triplets which satisfy the pythagoras’ theorem and are commonly, called pythagorean triplet.
 For example: 3, 4, 5;    5, 12, 13;    24, 10, 26;    24, 7, 25;    15, 8, 17

Appolonious Theorem:  



In triangle ABC, AD is median, which divides BC into two equal parts. Then, 
 AB2+AC2=2(AD2+BD2)=2(AD2+DC2)

Stewart theorem:

In Triangle ABC, AD divides side BC in the ratio m and n. (Here AD need not be median) then, 
m.b2+n.c2=a(d2+mn)

Mean proportionality and Mid Point theorem:



In the first triangle, DE // BC so AD / DB = AE / EC
In the second triangle, D and E are mid points of AB and AC respectively. Which implies,
AD / DB = AE / EC = 1
Also, DE = 1 / 2 BC

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