Important theorems of GEOMETRY by ABHISHEK JAIN Angles:An angle is a figure formed by two rays with a common initial point, say O. This point is called the vertex Types of Angles:1) A right angle is an angle of 900. e.g. Angle AOB
2) If an angle is less than 900, it is called acute. 3) If an angle is greater than 900 but less than 1800,it is called obtuse. 4) If an angle is of 1800 , it is called a straight angle, an angle greater than 1800 but less than 3600 is called a reflex angle. 5) Two angles whose sum is 1800 are called supplementary angles, each one is a supplement of the other. 6) Two angles whose sum is 900 are called complementary angles, each one is a complement of the other. 7) Two adjacent angles whose sum is 1800 are the angles of a linear pair. Angles and Intersecting lines: When two lines intersect, two pairs of vertically opposite angles are formed. Vertically opposite angles are equal. Thus, are equal. a b are equal.
Also, sum of all the angles at a point = 3600 i.e.
Angles and Parallel Lines: If a transversal (cutting line) cuts two parallel lines corresponding angles are equal i.e. a = e, d = f, b = h, c = g. Alternate angles are equal i.e. c = f, d = e. Interior angles on the same side of the transversal are supplementary, i.e. c + e = d + f = 180°
Triangles:A triangle is a polygon with three vertices and three sides (edges). It has three internal angles. When we add all the internal angles together, we will definitely get 180°. 1. Acute triangle: The triangle which has all acute angles (i.e less than 900). 2. Obtuse triangle: The triangle which has one obtuse angle (i.e greater that 900) 3. Right angle triangle: The triangle which has one right angle i.e 900.
4. Scalene triangle: A scalene triangle is a triangle that has no equal sides 5. Isosceles triangle: An isosceles triangle is a triangle that has two equal sides. 6. Equilateral triangle: The triangle of which all sides are equal is known as equilateral triangle & all angles are equal to 600.
Properties of Triangles 1. Sum of the angles of a triangle are 180° (angles of a triangle are supplementary)
2. The exterior angle of a triangle is equal to the sum of the interior opposite angles.
Here, 1 = 2 + 3 3. Angles opposite to two equal sides of a triangle are equal & vice versa.
Here, 1 = 2 4. If two sides of a triangle are unequal then the greater angle is opposite to greater side & vice versa
5. Two triangles are congruent if two angles & included side of one triangle is equal to the corresponding two angles & included side of the other triangles.
1= 3 or ABC
2 = 4, BC = EF DEF
6. If two sides & included angle of a triangle are equal to corresponding two sides & included angle of another triangle then the two triangles are congruent.
AB = DE, AC = DF , 1 = 2 ABC DEF 7. If two angles & non included side of one triangle are equal to corresponding two angles & non included side of another triangle then the two triangles are congruent.
8. If three sides of a triangle are equal to three sides of another triangle each to each then the triangles are congruent
AB = DE, AC = DF, BC = EF
9. Triangles on the same base & between the same parallels are equal in Area
Here, area of ΔABC = area of ΔBDC if AD is parallel to BC 10. Two right triangles are congruent if the hypotenuse & one side of one triangle are respectively equal to hypotenuse & one side of the other triangle.
AB = DE, AC = DF or ABC DEF 11. Sum of any two sides of a triangle is greater than the third. Similar Triangles 1. If a line is drawn parallel to one side of a triangle the other two sides are divided proportionally & vice versa
if DE || BC so if so
=
=
then DE || BC
2. If two triangles are equiangular (i.e. the corresponding angles are equal) then the triangles are similar & hence their sides are proportional
1 4, 2 5, 3 6,
=
=
3. If the corresponding sides of two triangles are proportional then the triangles are equiangular / similar. 4. Ratio of areas of two similar triangles is equal to the ratio of the squares of two corresponding sides. 5. In a right triangle the square of the hypotenuse is equal to the sum of the squares on the other two sides
6. In the given figure (obtuse angled triangle)
= if B > 90°
+ 2AB.BD
7. In the figure (Acute angles triangle)
= + if B < 90°
- 2BC.BD
Area of triangle 1) When lengths of the sides are given :-
Area = √ (
)(
)(
)
where, semi perimeter (s) =
2) When lengths of the base and altitude (height) are given:-
Area =
3) When lengths of two sides and the included angle are given:-
Area = 4) For Equilateral Triangle:-
Area =
√
5) For Isosceles Triangle
Area =
√4
6) When three median are given The area of a triangle can be expressed in terms of the medians by: A= √ (
)(
)(
)
Where, S= Apollonius Theorem
If AD is the median, then: AB² + AC² = 2(AD² + BD²)
Angle Bisector Theorem
If AD is the angle bisector for angle A, then:-
=
Inradius and circumradius of triangle # Inradius In case of Equalilateral triangle =
In case of right angle triangle =
In case of other triangles =
# Circumradius
In case of equilateral triangle =
In case of right angle triangle = In case of other triangles =
√
√
Circles:-
Circle illustration with circumference (C) , diameter (D) , radius (R) , and centre or origin (O)
Arc: any connected part of the circle. Centre: the point equidistant from the points on the circle. Chord: a line segment whose endpoints lie on the circle. Circular sector: a region bounded by two radii and an arc lying between the radii. Circular segment: a region, not containing the centre, bounded by a chord and an arc lying between the chord's endpoints. Circumference: the length of one circuit along the circle. Diameter: a line segment whose endpoints lie on the circle and which passes through the centre; or the length of such a line segment, which is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord, and it is twice the radius. Radius: a line segment joining the centre of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter. Secant: an extended chord, a coplanar straight line cutting the circle at two points. Semicircle: a region bounded by a diameter and an arc lying between the diameter's endpoints. It is a special case of a circular segment, namely the largest one. Tangent: a straight line that touches the boundary of circle at a single point.
Important result
Circumference = 2 Area = Area of semi circle = Circumference of semi circle = Length of arc ( ) = 2 Area of sector =
( (
°
°
2
)
)
Properties of circle 1. The perpendicular from the centre of a circle to a chord bisects the chord.
O is centre, AB is chord & OP is perpendicular to AB, AP = PB 2. Perpendicular bisectors of two chords of a circle passes through its centre (i.e. intersect at centre).
3. If two chords of a circle drawn from the same point are equal then the line bisecting the angle between them passes through centre (or is the diameter)
AB = BC, 1 = 2 then BP is diameter, centre lies on BP 4. Equal chords of a circle are equidistant from the centre & vice versa.
AB = CD or OP = OQ (O is the centre of the circle) 5. Angles in the same segment of a circle are equal
1 = 2 (being in the same segment) 6. Angle in a semicircle is a right angle
AB is the diameter 1 = 2 = 900
7. Angle which an arc subtends at the centre is double the angle subtended by the same arc at any other part of the circumference.
Arc AB subtends 1 at centre & 2 & 3 at circumference 1 22 23 8. From the above we come to know that 2 = 3 hence angles in the same segment of a circle are equal. 9. Equal chords of a circle subtend equal angles at the centre
Chord AB = Chord CD (O is the centre of the circle) 10. Opposite angles of a cyclic quadrilateral are supplementary
ABCD is cyclic quadrilateral then 1+ 2 = 3 + 4 = 1800
11. A tangent to a circle is perpendicular to the radius through the point of contact
AB is tangent to the circle at P with centre O
12. The length of two tangents from an external point are equal
OA = OB 13. If two chords of a circle intersect inside or outside the circle when produced, the rectangle formed by the two segments of one chord is equal in area to the rectangle formed by the two segments of the other.
Circle with centre O chords AB & CD intersect at P (fig. 1 inside & fig. 2 outside) PA.PB = PC.PD & if PT is tangent then PT2= PA.PB 14. If a chord is drawn through the point of contact of a tangent to a circle, then the angles which this chord makes with the given tangent are equal respectively to the angles formed in the corresponding alternate segments.
PQ is tangent to circle through pt. A & AB is a chord & 1= 2& 3= 4
Common Tangent To A Pair Of Circles Common tangents are lines or segments that are tangent to more than one circle at the same time. The possibility of common tangents is closely linked to the mutual position of circles. 1. If two circles touch inside, the two internal tangents vanish and the two external ones become a single tangent.
2. If two circles intersect, the common tangent is replaced by a common secant, whence there are only two external tangents.
3. If two circles touch each other outside, the two internal tangents coincide in a common tangent, thus there are three common tangents.
4. If two circles are separate, there are four common tangents, two inside and two outside.
Length of common tangent. If r1, r2 are the radii of two circles and d is distance between their centers, then (i) the length of a direct common tangent = √
(
)
(ii) the length of a transverse common tangent =√ ( ) Length of the tangent of two circles which touch each other at an external point with radius r1 & r2, then the length of direct common tangent = √
Quadrilaterals:Quadrilateral Square
Shape
Properties 1. All sides are equal 2. all angles are 90° 3.Diagonals are equal and Bisect each other at 90°
Important Results Area = a² = × D² (D = diagonal) Perimeter = 4a Diagonal = √2
Rectangle
Rhombus
Parallelogram
Trapezium
1. Opposite sides are equal and parallel 2. All angles are 90° 3. Diagonals are equal and bisect each other
Area =
1. All sides are equal and opposite sides are parallel 2. Opposite angles are equal 3. Diagonals are not equal 4. Diagonals bisect each other at 90°
Area =
1. opposite sides are parallel and equal 2. Opposite angles are equal 3. Diagonals of parallelogram bisect each other
Area = base × height Or b × h
1. only one pair of opposite sides are parallel 2. The diagonals cut the quadrilateral into four triangles of which one opposite pair are similar
Area = (sum of parallel sides) (height)
Perimeter = 2
2
Diagonal = √
Perimeter = sum of all sides 2
side² = d1 d 2 2 2
2
Perimeter = sum of all sides
If lengths of one diagonal and two offsets are given of any quadrilateral
Area =
(
)
If lengths of two diagonals and the included angle are given of any quadrilateral
Area =
Solids:Figure
Shape
Volume
CSA/LSA
4
Cube
)
2(
Cuboids
2(
2
Cylinder
Cone
Total surface area 6
2
1 3
)
(
( = √
Sphere
4 3
4
4
Hemisphere
2 3
2
3
Right prism
2
)
)
Right pyramid
1 3
1 2
Polygon:A closed plane figure made up of several line segments that are joined together. The sides do not cross each other. Exactly two sides meet at every vertex.
Side - one of the line segments that make up the polygon. Vertex - point where two sides meet. Two or more of these points are called vertices. Diagonal - a line connecting two vertices that isn't a side. Interior Angle - Angle formed by two adjacent sides inside the polygon. Exterior Angle - Angle formed by two adjacent sides outside the polygon. Sum of interior angle and exterior angle of any polygon is equal to 180° Types of Polygons Regular - all angles are equal and all sides are the same length. Regular polygons are both equiangular and equilateral. Equiangular - all angles are equal. Equilateral - all sides are the same length.
Convex - a straight line drawn through a convex polygon crosses at most two sides. Every interior angle is less than 180°.
Concave - you can draw at least one straight line through a concave polygon that crosses more than two sides. At least one interior angle is more than 180°.
Number of sides N 3 4 5
Name N – Gon Triangle Quadrilateral Pentagon
Exterior angle of any polygon
=
= Number of sides
Where
Interior angle of any polygon
= (1 0
= Number of sides Number of diagonals (
3) 2
= Number of sides
Number of triangles in any polygon (
Number of sides 6 7 8 10
4)( 6
5)
= Number of sides
)
Name Hexagon Heptagon Octagon Decagon
