One of the hardest sections to crack without preparation and one of the easiest with preparation. With so many formulas to learn and remember, this section is going to take a lot of time to master. Remember, read a formula, try to visualize the formula and solve as many questions related to the formula as you can. Knowing a formula and knowing when to apply it are two different abilities. The first will come through reading the formulae list and theory but the latter can come only through solving many different problems.
Basics of Parallel Lines and their Angles:
Equations of line:
Consider parallel lines AB, CD and EF as shown in the figure.
XY and MN are known as transversals
$$\angle $$ XPQ = $$\angle $$ PRS = $$\angle $$ RTU as corresponding angles are equal
Interior angles on the side of the transversal are supplementary. i.e. $$\angle $$ PQS + $$\angle $$ QSR = 180 degrees
Exterior angles on the same side of the transversal are supplementary. i.e. $$\angle $$ MQB + $$\angle $$ DSU = 180 degrees.
Two transversals are cut by three parallel lines in the same ratio i.e. PR / RT = QS / SU
Triangles
An angle less than 90 degrees is called an acute angle. An angle greater than 90 degrees is called an Obtuse angle.
A triangle with all sides unequal is called a Scalene triangle.
Nomenclature
A line joining the midpoint of a side with the opposite vertex is called a Median.
A perpendicular drawn from a vertex to the opposite side is called the Altitude.
The point of intersection of the three medians is the Centroid.
A triangle with two sides equal is called an Isosceles triangle. The two angles of an isosceles triangle that are not contained between the equal sides are equal.
A triangle with all sides equal is called an equilateral triangle. All angles of an equilateral triangle equal 60 degrees.
If x is the side of an equilateral triangle, then the
Altitude (h) =$$\frac{\sqrt{\ 3}}{2}x$$
Area =$$\frac{\sqrt{\ 3}}{4}x^2$$
Inradius = $$\frac{1}{3}\times\ h$$
Circumradius = $$\frac{2}{3}\times\ h$$
▪ Area of an isosceles triangle =$$\frac{a}{4}\sqrt{\ 4c^2-a^2}$$ (where a, b and c are the length of the sides of BC, AC and AB respectively b = c)
If a, b, and c are the lengths of the right-angled triangles, with 'c' being the hypotenuse, then
Circumradius = $$\dfrac{h}{2}$$
Inradius = $$\dfrac{\left(a+b-c\right)}{2}$$
Special triangles :
30, 60 and 90 degrees
45, 45 and 90 degrees
Area of a triangle A
Mid-Point Theorem: The line joining the midpoint of any two sides in a triangle is parallel to the third side and is half the length of the third side. If X is the midpoint of CA and Y is the midpoint of CB, then XY will be parallel to AB and XY = ½ * AB
Basic proportionality theorem: If a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points then it divides the two sides in the ratio of respective sides. If in a triangle ABC, D and E are the points lying on AB and BC respectively and DE is parallel to AC then AD/DB = EC/BE
Interior Angular Bisector theorem: In a triangle, the angular bisector of an angle divides the side opposite to the angle, in the ratio of the remaining two sides. In a triangle ABC if AD is the angle bisector of angle A then AD divides the side BC in the same ratio as the other two sides of the triangle. i.e. BD/ CD= AB/AC.
Exterior Angular Bisector theorem: The angular bisector of the exterior angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle. In a triangle ABC, if CE is the angular bisector of exterior
angle BCD of a triangle, then AE/BE = AC/BC
Similarity and Congruence:
Similar triangles :
If two triangles are similar then their corresponding angles are equal and the corresponding sides will be in proportion.
For any two similar triangles :
▪ Ratio of sides = Ratio of medians = Ratio of heights = Ratio of circumradii = Ratio of Angular bisectors.
▪ Ratio of areas = Ratio of the square of the sides.
Tests of similarity : (AA / SSS / SAS)
Congruent triangles
If two triangles are congruent then their corresponding angles and their corresponding sides are equal.
Tests of congruence : (SSS / SAS / AAS / ASA)
Important Points to Remember:
Circle Formulae
1. Area: $$\pi\ r^2$$
2. Circumference: $$2\pi r$$
3. Length of Arc: $$\left(\frac{\theta}{360}\right)\left(2\pi r\right)$$
4. Area of Sector: $$\left(\frac{\theta}{360}\right)\left(\pi r^2\right)$$
Inscribed angle Theorem : The angle inscribed by the two points lying on the circle, at the centre of the circle is twice the angle inscribed at any point on the circle by the same points:
Angles subtended by the same segment on the circle will be equal. So here angles a and b will be equal:
Equal Chords subtend equal angles at the centre
The angle subtended by a diameter on the circumference is a right angle
Tangents:
Direct common tangent: $$PQ^2=RS^2=D^2-\left(r_1-r_2\right)^2$$, where D is the distance between the centres:
Transverse common tangent: $$PQ^2=RS^2=D^2-\left(r_1+r_2\right)^2$$, where D is the distance between the centres:
If a quadrilateral has all its vertices on the circle and its opposite angles are supplementary (here x+y = 180^{0}), then that quadrilateral is called a cyclic quadrilateral.
Polygons:
▪ If all sides and all angles are equal, then the polygon is a regular polygon
▪ A regular polygon of n sides has n(n-3)/2 diagonals
▪ In a regular polygon of n sides, each exterior angle is 360/n degrees.
▪ Sum of measure of all the interior angles of a regular polygon is 180 (n-2)
degrees (where n is the number of sides of the polygon)
▪ Sum of measure of all the exterior angles of regular polygon is 360 degrees
ABCDEF is a regular hexagon with each side equal to ‘x’ then
▪ Each interior angle = 120 degrees
▪ Each exterior angle = 60 degrees
▪ Sum of all the exterior angles = 360 degrees
▪ Sum of all the interior angles = 720 degrees
▪ Area = $$\frac{3\sqrt{\ 3}}{2}a^2$$.
- Quadrilateral with only one pair of opposite sides are parallel.
- Area = $$\dfrac{1}{2}\times\ height\ \times\ \left(Sum\ of\ parallel\ sides\right)$$ i.e. If h is the height and AB, CD are the opposite parallel sides, then Area = $$\dfrac{1}{2}\times h\times\left(AB+CD\right)$$
- A line joining the midpoint of a side with the opposite vertex is called a Median.
- The point of intersection of the three medians is the Centroid.
- A perpendicular drawn from a vertex to the opposite side is called the Altitude.
-The three perpendicular bisectors of a triangle meet at a point called the Circumcentre.
- A circle drawn from this point with the circumradius would pass through all the vertices of the triangle.
- The three angle bisectors of a triangle meet at a point called the incentre of a triangle.
- The incentre is equidistant from the three sides, and a circle drawn from this point with the inradius would touch all the sides of the triangle.
- The point of intersection of the three altitudes is the Orthocentre.
In a right-angled triangle, we consider an angle 'x'
Sin(x) = $$\dfrac{opposite}{hypotenuse}$$
Cos(x) = $$\dfrac{adjacent}{hypotenuse}$$
Tan(x) = $$\dfrac{opposite}{adjacent}$$
Trigonometric values for important angles: 0. 30, 45, 60, 90.
Sin(0) = 0, Sin(30) = 1/2, Sin(45) = 1/$$\sqrt{\ 2\pm\ }$$, sin(60) = $$\frac{\sqrt{\ 3}}{2}$$, sin(90) = 1
Cos(0) = 1, Cos(30) = $$\frac{\sqrt{\ 3}}{2}$$, cos(45) = 1/$$\sqrt{\ 2\pm\ }$$, cos(60) = Sin(30) = 1/2, cos(90) = 0
Tan(0) = 0, Tan(30) = $$\frac{1}{\sqrt{\ 3}}$$, Tan(45) = 1, Tan(60) = $$\sqrt{\ 3}$$, Tan(90) = infinity.
Sin(90+x) = cos(x)
Cos(90+x) = -sin(x)
sin(-x) = -sin(x)
cos(-x) = cos(x)
$$\sin^2\left(x\right)+\cos^2\left(x\right)=1$$
sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
If 'p' is the given perimeter,
The total number of triangles
= $$\dfrac{p^2}{48}$$ if p is even
= $$\dfrac{\left(p+3\right)^2}{48}$$ if p is odd
The number of scalene triangles
= $$\dfrac{\left(p-6\right)^2}{48}$$ if p is even
= $$\frac{\left(p-3\right)^2}{48}$$ if p is odd
In all the cases, the value obtained should be rounded to the nearest integer.
- The Sum of all angles in a triangle is 180 degrees.
- An angle less than 90 degrees is called an acute angle.
- An angle greater than 90 degrees is called an Obtuse angle.
- The Sum of any two sides of a triangle is always greater than the third side.
- A triangle with two sides equal is called an Isosceles triangle
- A triangle with all sides equal is called an equilateral triangle.
- A triangle with all sides unequal is called a scalene triangle.
Characteristics of the four quadrants:
TYPE OF SOLID | LATERAL S.A. | TOTAL S.A. |
---|---|---|
Cube (all sides equal to a) | 4a^{2} | 6a^{2} |
Cuboid (length l, breadth b, height h) | 2(l + b)h | 2(lb + bh + hl) |
Right Circular Cylinder (radius r, height h) | 2πrh | 2πr(h + r) |
Right Circular Cone (radius r, height h, slant height l) | πrl | πr(l + r) |
Cone frustum (radii $$r_1$$, $$r_2$$, height h and slant height l) | $$\pi(r_1+r_2)l$$ | $$\pi(r_1+r_2)l+(r_1^2+r_2^2)$$ |
Sphere (radius r) | 4πr^{2} | 4πr^{2} |
Solid Hemisphere (radius r) | 2πr^{2} | 3πr^{2} |
Hollow Hemisphere (radius r) | 2πr^{2} | 2πr^{2} |
Pyramid | $$\frac{1}{2}\times\ Perimeter\times\ Slant\ Height$$ | $$Curved\ Surface\ area+Base\ area$$ |
Prism | $$Perimeter\times\ Height$$ | $$Curved\ Surface\ area+2\times\ Base\ area$$ |
TYPE OF SOLID | VOLUME |
---|---|
Cube (all sides equal to a) | a^{3} |
Cuboid (length l, breadth b, height h) | lbh |
Right Circular Cylinder (radius r, height h) | πr^{2}h |
Right Circular Cone (radius r, height h, slant height l) | $$\dfrac{1}{3}$$π r^{2}h |
Cone frustum (radii $$r_1$$, $$r_2$$, height h and slant height l) | $$V=\dfrac{1}{3}\pi h(r_2^2+r_1r_2+r_1^2)$$ |
Sphere (radius r) | $$\dfrac{4}{3}$$πr^{3} |
Solid Hemisphere (radius r) | $$\dfrac{2}{3}$$πr^{3} ^{} |
Hollow Hemisphere (radius r) | $$\dfrac{2}{3}$$πr^{3} ^{} |
Pyramid | $$\dfrac{1}{3}\times\ Area\ of\ the\ base\times\ Height$$ |
Prism | $$Base\ Area\times\ Height$$ |
Volume of Tetrahedron of side 'a' = $$\dfrac{a^3}{6\sqrt{\ 2}}$$
Coordinate geometry formulae: