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EduRev's Tip: The ratio of intercepts formed by a transversal intersecting three parallel lines is equal to the ratio of corresponding intercepts formed by any other transversal.
⇒ a/b = c/d = e/f
Area of a triangle:
= ½ x Base x Height
= ½ x Product of sides x Sine of included angle
= here s is the semi perimeter [s = (a+b+c)/2 ]
= r x s [r is radius of incircle]
[R is radius of circumcircle]
EduRev's Tip:
Theorems
Mid Point Theorem: The line joining the midpoint of any two sides is parallel to the third side and is half the length of the third side.
Basic Proportionality Theorem: If DE  BC, then AD/DB = AE/EC
Apollonius’ Theorem: AB^{2} + AC^{2} = 2 (AD^{2} + BD^{2})
Interior Angle Bisector Theorem: AE/ED = BA/BD
Right Angled Triangle:
∆ABC ≈ ∆ ADB ≈ ∆ BDC
BD^{2} = AD x DC and AB x BC = BD X DC
Equilateral Triangle:
All angles are equal to 60°. All sides are equal also.
Height =
Area =
Inradius = 1/3 Height
Circumradius = 2/3 Height.
Isosceles Triangle:
Angles equal to opposite sides are equal.
Area
30°60°90° Triangle
Area
45°45°90° Triangle
Area = x^{2}/2
30°30°120° Triangle
Area =
Similarity of Triangles
Two triangles are similar if their corresponding angles are congruent and corresponding sides are in proportion.
Tests of similarity: (AA / SSS / SAS)
For similar triangles, if the sides are in the ratio of a:b
⇒ Corresponding heights are in the ratio of a:b
⇒ Corresponding medians are in the ratio of a:b
⇒ Circumradii are in the ratio of a:b
⇒ Inradii are in the ratio of a:b
⇒ Perimeters are in the ratio of a:b
⇒ Areas are in the ratio a^{2} : b^{2}
Congruency of Triangles
Two triangles are congruent if their corresponding sides and angles are congruent.
Tests of congruence: (SSS / SAS / AAS / ASA)
All ratios mentioned in similar triangle are now 1:1
Polygons
Sum of interior angles = (n2) x 180° = (2n4) x 90° Sum of exterior angles = 360°
Number of diagonals =
Number of triangles which can be formed by the vertices = ^{n}C_{3}
Regular Polygon:
If all sides and all angles are equal, it is a regular polygon. All regular polygons can be inscribed in or circumscribed about a circle.
Area = ½ x Perimeter x Inradius {Inradius is the perpendicular from centre to any side}
Each Interior Angle = Exterior = 360°/n
Quadrilaterals:
Sum of the interior angles = Sum of the exterior angles = 360°
Area for a quadrilateral is given by ½ d_{1} d_{2} Sinθ.
Cyclic Quadrilateral
If all vertices of a quadrilateral lie on the circumference of a circle, it is known as a cyclic quadrilateral.
Opposite angles are supplementary
Area = where s is the semi perimeter
EduRev's Tip:
Parallelogram
EduRev's Tip:
Rhombus
A parallelogram with all sides equal is a Rhombus. Its diagonals bisect at 90°.
Perimeter = 4a; Area = ½ d_{1} d_{2} ; Area = d x
Rectangle: A parallelogram with all angles equal (90°) is a Rectangle. Its diagonals are congruent. Perimeter = 2(l+b); Area = lb
Square: A parallelogram with sides equal and all angles equal is a square. Its diagonals are congruent and bisect at 90°.
Perimeter = 4a; Area = a^{2}; Diagonals = a√2
EduRev's Tip: From all quadrilaterals with a given area, the square has the least perimeter. For all quadrilaterals with a given perimeter, the square has the greatest area.
Kite
Trapezium / Trapezoid
A quadrilateral with exactly one pair of sides parallel is known as a Trapezoid. The parallel sides are known as bases and the nonparallel sides are known as lateral sides.
Area = ½ x (Sum of parallel sides) x Height
Median, the line joining the midpoints of lateral sides, is half the sum of parallel sides.
EduRev's Tip: Sum of the squares of the length of the diagonals = Sum of squares of lateral sides + 2 Product of bases.
⇒ AC^{2} + BD^{2} = AD^{2} + BC^{2} + 2 x AB x CD
Isosceles Trapezium
The nonparallel sides (lateral sides) are equal in length. Angles made by each parallel side with the lateral sides are equal.
EduRev's Tip: If a trapezium is inscribed in a circle, it has to be an isosceles trapezium. If a circle can be inscribed in a trapezium, Sum of parallel sides = Sum of lateral sides.
Hexagon (Regular)
Perimeter = 6a; Area =
Sum of Interior angles = 720°.
Each Interior Angle = 120°. Exterior = 60°
Number of diagonals = 9 {3 big and 6 small}
Length of big diagonals (3) = 2a
Length of small diagonals (6) = √3a
EduRev's Tip: A regular hexagon can be considered as a combination of six equilateral triangles. All regular polygons can be considered as a combination of ‘n’ isosceles triangles.
Area of a Pentagon = 1.72 a^{2}
Area of an Octagon = 2(√2 + 1) a^{2}
Circles
Diameter = 2r; Circumference = 2πr; Area = πr^{2}
Chords equidistant from the centre of a circle are equal.
A line from the centre, perpendicular to a chord, bisects the chord.
Equal chords subtend equal angles at the centre.
The diameter is the longest chord of a circle.
A chord /arc subtends equal angle at any point on the circumference and double of that at the centre.
Chords / Arcs of equal lengths subtend equal angles.
Chord AB divides the circle into two parts: Minor Arc AXB and Major Arc AYB.
Measure of arc AXB = ∠AOB = θ
Length (arc AXB) =
Area (sector OAXB) =
Area of Minor Segment = Shaded Area in above figure
⇒ Area of Sector OAXB  Area of ∆ OAB
Properties of Tangents, Secants and Chords
The radius and tangent are perpendicular to each other.
There can only be two tangents from an external point, which are equal in length PA = PB
PA x PB = PC x PD
θ = ½ [ m(Arc AC) – m(Arc BD) ]
PA x PB = PC x PD
θ = ½ [ m(Arc AC) + m(Arc BD) ]
Properties (contd.)
PA x PB = PC2
θ = ½ [ m(Arc AC)  m(Arc BC) ]
Alternate Segment Theorem
The angle made by the chord AB with the tangent at A (PQ) is equal to the angle that it subtends on the opposite side of the circumference.
⇒ ∠BAQ = ∠ACB
Common Tangents
⇒ AD = BC =
Length of the Transverse Common Tangent (TCT)
⇒ RT = SU =
EduRev's Tip:
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