More Tips on Geometry
-> If any parallelogram can be inscribed in a circle , it must be a rectangle.
-> If a trapezium can be inscribed in a circle it must be an isosceles trapezium (i:e oblique sides equal).
-> 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) .
-> Area of a regular hexagon : root(3)*3/2*(side)*(side)
-> 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.
-> In any triangle a/SinA = b/SinB =c/SinC=2R , where R is the circumradius
-> In an isosceles triangle , the perpendicular from the vertex to the base or the angular bisector from vertex to base bisects the base.
-> In any triangle the angular bisector of an angle bisects the base in the ratio of the other two sides.
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Cyclic Quadrilateral
-> For a cyclic quadrilateral , area = root( (s-a) * (s-b) * (s-c) * (s-d) ) , where s=(a+b+c+d)/2
-> For a cyclic quadrilateral , the measure of an external angle is equal to the measure of the internal opposite angle.
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Quadrilateral
-> If a quadrilateral circumscribes a circle , the sum of a pair of opposite sides is equal to the sum of the other pair . -> The quadrilateral formed by joining the angular bisectors of another quadrilateral is always a rectangle.
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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 .
In any triangle a=b*CosC + c*CosB b=c*CosA + a*CosC c=a*CosB + b*CosA
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Area of a hexagon
Area of a hexagon = root(3) * 3 * (side)^2
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Area of a trapezium
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.
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Co-ordinates of Centroidi
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) .
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Ratio of Radii of Corcumcircle & In-circle
The ratio of the radii of the circumcircle and incircle of an equilateral triangle is 2:1 .
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Area of Parallelogram
Area of a parallelogram = base * height
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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) .
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Similar Cones
For similar cones , ratio of radii = ratio of their bases. The HCF and LCM of two nos. are equal when they are equal .
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Pyramids
Volume of a pyramid = 1/3 * base area * height
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