Geometry
General Notions and useful shortcuts:
Polygons:
(1) For any regular polygon, the sum of the interior angles is equal to 360 degrees
(2) If any parallelogram can be inscribed in a circle , it must be a rectangle.
(2.1)Given the coordinates (a,b) (c,d) (e,f) (g,h) of a parallelogram, the coordinates of the meeting point of the diagonals can be found out by solving for [(a+e)/2, (b+f)/2] =[ (c+g)/2, (d+h)/2]
(3) If a trapezium can be inscribed in a circle it must be an isosceles trapezium (i:e oblique sies equal).
(4) For an isosceles trapezium , sum of a pair of opposite sides is equal in length to the sum of the other pair of opposite sides .(i:e AB+CD = AD+BC , taken in order) .
Triangles
(1) In an isosceles triangle , the perpendicular from the vertex to the base or the angular bisector from vertex to base bisects the base.
(2) In any triangle the angular bisector of an angle bisects the base in the ratio of the other two sides.
(3) The ratio of the radii of the circumcircle and incircle of an equilateral triangle is 2:1 .
(4.1)In any triangle
a=b*CosC + c*CosB
b=c*CosA + a*CosC
c=a*CosB + b*CosA
(4.2)In any triangle
a/SinA = b/SinB =c/SinC=2R , where R is the circumradius
cosC = (a^2 + b^2 - c^2)/2ab
sin2A = 2 sinA * cosA
cos2A = cos^2(A) - sin^2 (A)
(5.1)APPOLLONIUS THEOREM:
In a triangle , if AD be the median to the side BC , then
AB^2 + AC^2 = 2(AD^2 + BD^2) or 2(AD^2 + DC^2) .
(5.2) Appolonius theorem could be applied to the 4 triangles formed in a parallelogram.
(6) The coordinates of the centroid of a triangle with vertices (a,b) (c,d) (e,f)
is((a+c+e)/3 , (b+d+f)/3) .
(7) Let a be the side of an equilateral triangle . then if three circles be drawn
inside this triangle touching each other then each's radius = a/(2*(root(3)+1))
(8) Let W be any point inside a rectangle ABCD .
Then WD^2 + WB^2 = WC^2 + WA^2
(9) Some pythagorean triplets:
3,4,5 (3^2=4+5)
5,12,13 (5^2=12+13)
7,24,25 (7^2=24+25)
8,15,17 (8^2 / 2 = 15+17 )
9,40,41 (9^2=40+41)
11,60,61 (11^2=60+61)
12,35,37 (12^2 / 2 = 35+37)
16,63,65 (16^2 /2 = 63+65)
20,21,29(EXCEPTION)
Quadrilateral
(1) For a cyclic quadrilateral , the measure of an external angle is equal to the measure of the
internal opposite angle.
(2) If a quadrilateral circumscribes a circle , the sum of a pair of opposite sides is equal
to the sum of the other pair .
(3) the quadrilateral formed by joining the angular bisectors of another quadrilateral is
always a rectangle.
Areas:
(1)Area of a triangle
1/2*base*altitude = 1/2*a*b*sinC = 1/2*b*c*sinA = 1/2*c*a*sinB = root(s*(s-a)*(s-b)*(s-c))
where s=a+b+c/2
=a*b*c/(4*R) where R is the CIRCUMRADIUS of the triangle = r*s ,where r is the inradius of the
triangle
(2.1) For a cyclic quadrilateral , area = root( (s-a) * (s-b) * (s-c) * (s-d) ) , where s=(a+b+c+d)/2
(2.2) For any quadrilateral whose diagonals intersect at right angles , the area of the quadrilateral is 0.5*d1*d2, where d1,d2 are the lenghts of the diagonals.
(3.1) Area of a regular hexagon : root(3)*3/2*(side)*(side)
(3.2) Area of a hexagon = root(3) * 3 * (side)^2
(4) Area of a parallelogram = base * height
(5) Area of a trapezium = 1/2 * (sum of parallel sids) * height = median * height
where median is the line joining the midpoints of the oblique sides.
Stereometry
(1) for similar cones , ratio of radii = ratio of their bases.
(2) Volume of a pyramid = 1/3 * base area * height
Number properties
(1) Product of any two numbers = Product of their HCF and LCM .
Hence product of two numbers = LCM of the numbers if they are prime to each other .
(2) The HCF and LCM of two nos. are equal when they are equal .
(3) For any 2 numbers a>b
a>AM>GM>HM>b (where AM, GM ,HM stand for arithmetic, geometric , harmonic menasa
respectively)
(4) (GM)^2 = AM * HM
(5) For three positive numbers a, b ,c
(a+b+c) * (1/a+1/b+1/c)>=9
(6) For any positive integer n
2<= (1+1/n)^n <=3
(7) a^2+b^2+c^2 >= ab+bc+ca
If a=b=c , then the equality holds in the above.
(8) a^4+b^4+c^4+d^4 >=4abcd
(9) If a+b+c+d=constant , then the product a^p * b^q * c^r * d^s will be maximum
if a/p = b/q = c/r = d/s
(10) (m+n)! is divisible by m! * n! .
(11.1)If n is even , n(n+1)(n+2) is divisible by 24
(11.2)If n is any integer , n^2 + 4 is not divisible by 4
(12) x^n -a^n = (x-a)(x^(n-1) + x^(n-2) + .......+ a^(n-1) ) ......Very useful for finding
multiples .For example (17-14=3 will be a multiple of 17^3 - 14^3)
(13) when a three digit number is reversed and the difference of these two
numbers is taken , the middle number is always 9 and the sum of the other two
numbers is always 9 .
(14) Let 'x' be certain base in which the representation of a number is 'abcd' , then
the decimal value of this number is a*x^3 + b*x^2 + c*x + d
(15) 2<= (1+1/n)^n <=3
(16) (1+x)^n ~ (1+nx) if x<<<1
(17) |a|+|b| = |a+b| if a*b>=0 else |a|+|b| >= |a+b|
(18) In a GP (Geometric Progression?) the product of any two terms equidistant from a term is always constant .
(19)The sum of an infinite GP = a/(1-r) , where a and r are resp. the first term and common ratio of the GP .
(20)If a1/b1 = a2/b2 = a3/b3 = .............. , then each ratio is equal to
(k1*a1+ k2*a2+k3*a3+..............) / (k1*b1+ k2*b2+k3*b3+..............) , which is also equal to
(a1+a2+a3+............./b1+b2+b3+..........)
General Notions and useful shortcuts:
Polygons:
(1) For any regular polygon, the sum of the interior angles is equal to 360 degrees
(2) If any parallelogram can be inscribed in a circle , it must be a rectangle.
(2.1)Given the coordinates (a,b) (c,d) (e,f) (g,h) of a parallelogram, the coordinates of the meeting point of the diagonals can be found out by solving for [(a+e)/2, (b+f)/2] =[ (c+g)/2, (d+h)/2]
(3) If a trapezium can be inscribed in a circle it must be an isosceles trapezium (i:e oblique sies equal).
(4) For an isosceles trapezium , sum of a pair of opposite sides is equal in length to the sum of the other pair of opposite sides .(i:e AB+CD = AD+BC , taken in order) .
Triangles
(1) In an isosceles triangle , the perpendicular from the vertex to the base or the angular bisector from vertex to base bisects the base.
(2) In any triangle the angular bisector of an angle bisects the base in the ratio of the other two sides.
(3) The ratio of the radii of the circumcircle and incircle of an equilateral triangle is 2:1 .
(4.1)In any triangle
a=b*CosC + c*CosB
b=c*CosA + a*CosC
c=a*CosB + b*CosA
(4.2)In any triangle
a/SinA = b/SinB =c/SinC=2R , where R is the circumradius
cosC = (a^2 + b^2 - c^2)/2ab
sin2A = 2 sinA * cosA
cos2A = cos^2(A) - sin^2 (A)
(5.1)APPOLLONIUS THEOREM:
In a triangle , if AD be the median to the side BC , then
AB^2 + AC^2 = 2(AD^2 + BD^2) or 2(AD^2 + DC^2) .
(5.2) Appolonius theorem could be applied to the 4 triangles formed in a parallelogram.
(6) The coordinates of the centroid of a triangle with vertices (a,b) (c,d) (e,f)
is((a+c+e)/3 , (b+d+f)/3) .
(7) Let a be the side of an equilateral triangle . then if three circles be drawn
inside this triangle touching each other then each's radius = a/(2*(root(3)+1))
(8) Let W be any point inside a rectangle ABCD .
Then WD^2 + WB^2 = WC^2 + WA^2
(9) Some pythagorean triplets:
3,4,5 (3^2=4+5)
5,12,13 (5^2=12+13)
7,24,25 (7^2=24+25)
8,15,17 (8^2 / 2 = 15+17 )
9,40,41 (9^2=40+41)
11,60,61 (11^2=60+61)
12,35,37 (12^2 / 2 = 35+37)
16,63,65 (16^2 /2 = 63+65)
20,21,29(EXCEPTION)
Quadrilateral
(1) For a cyclic quadrilateral , the measure of an external angle is equal to the measure of the
internal opposite angle.
(2) If a quadrilateral circumscribes a circle , the sum of a pair of opposite sides is equal
to the sum of the other pair .
(3) the quadrilateral formed by joining the angular bisectors of another quadrilateral is
always a rectangle.
Areas:
(1)Area of a triangle
1/2*base*altitude = 1/2*a*b*sinC = 1/2*b*c*sinA = 1/2*c*a*sinB = root(s*(s-a)*(s-b)*(s-c))
where s=a+b+c/2
=a*b*c/(4*R) where R is the CIRCUMRADIUS of the triangle = r*s ,where r is the inradius of the
triangle
(2.1) For a cyclic quadrilateral , area = root( (s-a) * (s-b) * (s-c) * (s-d) ) , where s=(a+b+c+d)/2
(2.2) For any quadrilateral whose diagonals intersect at right angles , the area of the quadrilateral is 0.5*d1*d2, where d1,d2 are the lenghts of the diagonals.
(3.1) Area of a regular hexagon : root(3)*3/2*(side)*(side)
(3.2) Area of a hexagon = root(3) * 3 * (side)^2
(4) Area of a parallelogram = base * height
(5) Area of a trapezium = 1/2 * (sum of parallel sids) * height = median * height
where median is the line joining the midpoints of the oblique sides.
Stereometry
(1) for similar cones , ratio of radii = ratio of their bases.
(2) Volume of a pyramid = 1/3 * base area * height
Number properties
(1) Product of any two numbers = Product of their HCF and LCM .
Hence product of two numbers = LCM of the numbers if they are prime to each other .
(2) The HCF and LCM of two nos. are equal when they are equal .
(3) For any 2 numbers a>b
a>AM>GM>HM>b (where AM, GM ,HM stand for arithmetic, geometric , harmonic menasa
respectively)
(4) (GM)^2 = AM * HM
(5) For three positive numbers a, b ,c
(a+b+c) * (1/a+1/b+1/c)>=9
(6) For any positive integer n
2<= (1+1/n)^n <=3
(7) a^2+b^2+c^2 >= ab+bc+ca
If a=b=c , then the equality holds in the above.
(8) a^4+b^4+c^4+d^4 >=4abcd
(9) If a+b+c+d=constant , then the product a^p * b^q * c^r * d^s will be maximum
if a/p = b/q = c/r = d/s
(10) (m+n)! is divisible by m! * n! .
(11.1)If n is even , n(n+1)(n+2) is divisible by 24
(11.2)If n is any integer , n^2 + 4 is not divisible by 4
(12) x^n -a^n = (x-a)(x^(n-1) + x^(n-2) + .......+ a^(n-1) ) ......Very useful for finding
multiples .For example (17-14=3 will be a multiple of 17^3 - 14^3)
(13) when a three digit number is reversed and the difference of these two
numbers is taken , the middle number is always 9 and the sum of the other two
numbers is always 9 .
(14) Let 'x' be certain base in which the representation of a number is 'abcd' , then
the decimal value of this number is a*x^3 + b*x^2 + c*x + d
(15) 2<= (1+1/n)^n <=3
(16) (1+x)^n ~ (1+nx) if x<<<1
(17) |a|+|b| = |a+b| if a*b>=0 else |a|+|b| >= |a+b|
(18) In a GP (Geometric Progression?) the product of any two terms equidistant from a term is always constant .
(19)The sum of an infinite GP = a/(1-r) , where a and r are resp. the first term and common ratio of the GP .
(20)If a1/b1 = a2/b2 = a3/b3 = .............. , then each ratio is equal to
(k1*a1+ k2*a2+k3*a3+..............) / (k1*b1+ k2*b2+k3*b3+..............) , which is also equal to
(a1+a2+a3+............./b1+b2+b3+..........)
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