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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