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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