Geometry has some major theorems. One should be clear about
them, the ones on similarity of triangles, congruency of triangles,
pythagoras, area and volume formula. Kindly refer to a text book
for revising such concepts, I would recommend to go through
NCERT books (from fifth standard to tenth
standard). Anyway let's look at an important concept here!

**The major theorems which we always need are
:**

**Theorem 1: Pythagoras Theorem :** where are sides of a right
angled triangle.

Clearly, C is the largest side, we call it hypotenuse.

The triplets of real numbers (a,b,c) which satisfy the above
theorem is called pythagorean triplets. They are of real interest
in all kinds of work.

**Example 1: The length of one of the legs of a right
triangle exceeds the length of the other leg by 10 cm but is
smaller than that of the hypotenuse by 10 cm. Find the
hypotenuse.**

The obvious solution is ( I have jumped a step)

solving we have ( a can't be zero, its side of a
triangle)%

hypo is

P.s : we have avoided the cumbersome assumption of sides as
and

Tipster clue: See this, the smallest integer Pythagorean triplet
is (3,4,5) so all numbers of the form (3k,4k,5k) will be
Pythagorean!

**Practice Problem 1:** Find the sum of the lengths
of the sides of a right angled triangle if the Circumradius=15 and
inradius=6

**Theorem 2: Sin law**

where a,b,c are sides opposite and respectively
and R is circumradius of Triangle ABC.

Very useful theorem, though we have entered the domain of
trigonometry, but trigonometry, plane geometry and coordinate
geometry are very important for each other to co exist.

**Theorem 3: Cosine law**

( the notations remain the same as Theorem
2). The theorem can be similarly used for other angles too.

**Practice Problem 2:** Find the angle between the
diagonal of a rectangle with perimeter 2p and area

**Example 2: Find the length of the base of an isosceles
triangle with area S and vertical angle A.**

How do we start with this, we can off course going to need some
basic geometry knowledge. let me tell you all of it. First the
vertical angle of an isosceles triangle is the angle between the
two equal sides( unless otherwise mentioned). The Perpendicular
dropped on the unequal side from the opposite vertex, bisects the
vertical angle as well as bisects the side. It means if we have a
triangle ABC with and perpendicular to
then
and .

The last thing we need is that area of a triangle is or
for an isosceles triangle as

now given

Now as AD bisects the vertical angle and then use

hence

we can put the value of b from (1) and we are done !

**Example 3: In Triangle ABC, AD,BE
and CF are the medians which intersect at G. ABCH is trapezium with AH=5units , and BC=10units and
Area( Tr BHC)=35 Sq units. Find the ratio
of Area( BDFG): Area( ABCH). ( note we have H and C on same side of B
)**

Here we again need to know this. The three medians divide the
triangle into three triangle of equal area . Also they divide it
into three quadrilaterals of equal area. So

Next comes, the traingles drawn on the same base and between
same parallel lines have equal area. Hence as we know the base BC, we know the
altitude

so our ratio is

**Image Credit:** billjacobus1

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