Tips on Geometry Questions for MP State Exams

How to Prepare Geometry?

Questions from Geometry in the mathematics section of the state exam are asked from following sub-topics:

Before we start, please note that we have not covered the basic properties of the above-mentioned figures.

Here are the must-know properties of triangles:

The exterior angle of a triangle is equal to the sum of interior opposite angles.
Sum of the length of 2 sides of a triangle is greater than the 3^rd
Difference between the length of 2 sides of a triangle is less than the 3^rd

Centroid

It is the point where all the 3 medians of the triangle meet. In the figure below, O is the centroid, and,
BD = CD, AE = EC and AF = FB
AO:OD = 2:1 (The same stands true for CO:OF & BO:OE)
AB² + AC² = 2 (AD² + BD²) {Apollonius theorem}
(The sum of squares of any two sides of any triangle = 2* (Square of ½ of the third side + Square on the median bisecting the third side).

Circumcentre

It’s the meeting point of the perpendicular bisectors of the sides of a triangle. For triangle ABC, O is the circumcentre.
The circle shown above is the circumcircle and has its centre at O. Its radius is called circum-radius.
AB*BC*AC = 4*R* Area of the triangle. (where R = circum-radius)

Orthocentre

It’s the point where the perpendiculars to each side of the triangle meet.
Points B, C, E, and F lie on the circumference of a circle & form a cyclic quadrilateral.
Centroid divides the line joining the orthocentre & circumcentre in the ratio of 2:1.

Incentre

It’s the point where the angle bisectors of a triangle meet.
(Incentre is the centre of the incircle of a triangle)
Incentre is equidistant from all the sides of the triangle.
BD / DC = AB / AC
Angle BOC = 90^o + ½ of Angle BAC
Inradius is the radius of Incircle. Its value is:
(a) Equilateral triangle = side/ (2√3)
(b) Right-angled triangle = (sum of perpendicular sides – Hypotenuse)/2
(c) Other triangles = (Area of the triangle)/ (Semi-perimeter of the triangle)

Here are some important properties of Circles:

	A perpendicular drawn from the centre of a circle to the chord bisects it Angle OXB = 90^o So, AX = XB
	Angle AOC = 2* Angle ABC
	For chords intersecting internally or externally: PA * PB = PC * PD
	PT is the tangent & PAB is the secant, then: PA * PB = PT²
	XY is the common tangent for 2 circles, then, length of XY: = √ {OO’)² – (R₁ - R₂)²}
	XAY is the tangent to the given circle, then, Angle XAC = Angle ABC Angle YAB = Angle ACB

Equation of a line: ax + by + c = 0
Slope-Intercept Equation: y = mx + c
(m = slope of the line, c = intercept on y-axis)
Equation of line with slope ‘m’, passing through (x₁, y₁)
y – y₁ = m (x – x₁)
Slope of a line = (y₂ – y₁)/ (x₂ – x₁) = - (coefficient of x/ coefficient of y)
If ‘θ’ is the angle between 2 lines with slopes m₁ & m₂, then,
Tan θ = (m₂ – m₁)/ (1 + m₁m₂) or -(m₂ – m₁)/ (1 + m₁m₂)
Tan θ = 0, then lines are parallel,
Cot θ = 0, then lines are perpendicular.
Two lines parallel to each other may be represented as
ax + by + c₁ = 0
ax + by + c₂ = 0
Two lines perpendicular to each other may be represented as
ax + by + c₁ = 0
bx - ay + c₂ = 0
Coordinates of mid-point of a line formed by points (x₁, y₁) & (x₂, y₂) is given by:
(x₁ + x₂)/ 2, (y₁ + y₂)/2
Distance between two points, (x₁, y₁) & (x₂, y₂) is given by:
√ [(x₂ – x₁)² + (y₂ – y₁)²]
If points (x₁, y₁), (x₂, y₂) & (x₃, y₃) form a triangle, then its area is given by:
½ [x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁– y₂)]