CAT
GEOMETRY AND MENSURATION
Lines and angles
Sum of all angles made on a line, meeting at a point is 180
Sum of all angles made by number of lines meeting at single point is
360
Vertically opposite angles are equal
Line passing through a midpoint of another line is bisector of the line,
which divides the line in two equal parts
Angle bisector – line drawn at vertex and dividing the angle in 2 equal
parts. Any point on angle bisector is equidistant from other two arms
of angle.
Perpendicular bisector – if a line A is perpendicular to line B and it
passes through midpoint if B as well, then it is perpendicular bisector.
Any point on perpendicular bisector is equidistant from both ends of
the line segment.
In case of parallel lines cut by transversal:
The alternate and corresponding angles are equal
The sum of interior/exterior angles on same side respectively is 180
Basic proportionality theorem
If 3 or more parallel lines are intersected by 2 or more transversals,
then corresponding segments formed are in equal ratio.
TRAINGLES
Sum of all three angles is 180
Sum of exterior angles is 360
The exterior angle of a triangle is equal to two opposite interior angles.
Perpendicular bisector passes through the midpoint of a side and it is
not necessary it passes through the opposite vertex.
The perpendicular drawn from vertex to opposite side is the altitude of
the side.
The line joining the midpoint of a side to opposite vertex is called
median. A median divides a triangle into two equal halves in terms of
area of triangle.
Sum of any two sides is greater than third side and difference of any two
sides is lesser than third side
Greater the angle, greater the side opposite to it and vice versa.
There can only be one right or one obtuse angle and there can’t be both
a right and obtuse angle together in a same triangle.
In Iso. Triangle, altitude drawn to base is also perpendicular bisector of
base and angle bisector of vertical angle and is also the median.
In Eqi. Triangle, perpendicular bisector, median, and altitude coincide
The altitude in equilateral triangle is equal to √3a/2, where a = side
with the angle bisector of opposite vertex
The median drawn to the hypotenuse is equal to half of the length of the
hypotenuse.
Length of perpendicular on hypotenuse – original triangle area formula =
new triangle area formula (1/2 X base X height = ½ X base into height)
In right angled triangle, the square of hypotenuse = sum of squares of
other 2 sides
Angle bisector theorem:
Internal bisector of angle divides the opposite side in ratio of other 2 sides
Apollonius theorem:
If AD is median to side BC, then AB2 + AC2 = 2 (AD2 + BD2)
In acute/obtuse triangle, the square of side opposite to acute/opposite angle is
greater than sum of squares of other two sides by a qty equal to twice the
product of one of the sides containing acute/obtuse angle and the projection
of second side on first side. AD is median, AC2=AB2+BC2 -/+ 2BC.BD
Radii of circumcircle and incircle in eqi. Triangle is in ratio 2:1 and therefore
circumcircle and incircle’s area is in ratio 4:1.
When 3 medians are drawn, the resulting six triangles are equal in area and
each of them are equal to 1/6th of the area of original triangle.
GEOMETRIC CENTERS
Incenter (I)
meeting point of 3 internal angle
CIRCUMCENTER (S) bisectors
Meeting point of all 3 perepndicualr equidistant from all three sides i.e.
bisectors perpendicular drawn from three sides are
equidistant from all 3 vertices and is the equal and called inradius
radius Incircle touches all three sides
circumcircle passes through all three inradius is less than half of any altitude of
vertices triangle
geometric centers of
traingle CENTROID (G)
meeting point of 3 medians
ORTHOCENTER (O) divides median in ration 2:1, with
Meeting point of all 3 altitudes greater part towards vertex
ACUTE ANGLE OBTUSE ANGLE RIGHT ANGLE
.
.
TRAINGLE TRAINGLE TRIANGLE
circumcenter and circum and ortho circum on
othocenter inisde outside the midpoint of
traingle hyopoyenuse
ortho at vetrex
whre right angle
is formed
EULER’S LINE – in any triangle, except eqi. The ortho, centroid and
circum are collinear and lie on the same line called Euler’s line
The centroid divides ortho and circum internally in ratio 2:1
In eqi. All four centers coincide, i.e., they are at the same point.
The circumradius of an equilateral triangle is equal to (a / √3)
The inradius of an equilateral triangle is equal to a/2√3
In triangle abc, if I is incentre, then angle BIC = 90 +1/2 BAC
In triangle abc, if S is cicrumcentre, then angle BsC = 2BAC
The line joining midpoints of 2 sides, is parallel to 3 side and is half the
length of the third side
In triangle ABC, if O is ortho, and F and E are points on AB and AC where
altitude meets,then angle FOB=EOC=BAC and FOE = 180 - BAC
Congruency of triangles
All corresponding sides and angles are equal and size and shape are
same.
Perimeters and areas are equal
Tests for congruency - SSS, SAS, ASA, AAS and (RHS for right angle
triangle)
Similar triangles
corresponding angles are equal.
Same in shape but size is not similar necessarily
Corresponding sides are in equal ratio
Ratio of corresponding altitudes (heights), medians, angular bisectors,
inradii, circumradii, perimeters is equal to ratio of corresponding sides
Ratio of areas = ratio of squares of two corresponding sides
Tests for similarity – AA, SAS, SSS
BPT In triangles – a line drawn inside triangle parallel to one side divides
the other two sides in same proportion.
Midpoint theorem – segment joining the midpoint of any two sides is
parallel to third side and half the length of third side
Triplets of right-angle triangle
Triplets with odd number, let’s say, 3 is smallest side, then 32/2 = 4.5 and then
take integers below and above 4.5, therefore 3,4,5 is triplets
Triplets with odd number, let’s say, 8 is smallest side, now here do reverse,
first divide 8 by 2 and then take square, (8/2)2 = 16, and then take integers
below and above 16, therefore, 8,15,17 are triplets.
Commonly used triplets – 3,4,5 / 5,12,13 / 7,24,25 / 9,40,41 / 8,15,17
How to use triplets – if sides are 24,32,40, we don’t need to solve and find if
it’s a right-angle triangle, we take ratio, i.e., 3:4:5 and yes, it’s a triplet, hence it
is a right angle triangle, no need to solve
Similarly if sides are 39, 52 and we are asked to find third side, we notice 39 &
52 is 13X3 and 13X4 and then third side = 13X5, hence3:4:5 which is a triplet
