Tips on Geometry Questions for UP State Exams

By Nitin Singhal|Updated : January 13th, 2021

Geometry section of the various state exams usually accounts for 18-20 questions in the Mathematics section, with question carrying 1 mark. Thus, out of the total 100 marks of the Maths section, Geometry in the state exams has a significant weightage, and hence, it is important to have a thorough preparation of Geometry for the state exams.

Here are some important tips to prepare the geometry section of the state exams. Properties of the different figures and state exams geometry preparation tips, if studied properly, will definitely aid students in securing good marks from this section.

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How to Prepare Geometry?

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

Coordinate Geometry
Lines, Angles & their properties
Properties of Triangles, Circles, Rectangles, Squares, Polygons etc.

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

Triangles & their properties:

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)

Circles & their properties

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

Polygons:

Opposite angles are supplementary in case of cyclic quadrilaterals.
For any regular polygon, sum of all the exterior angles = 360^o
For a polygon of side n, value of each exterior angle = (360^o / n)^o
For a polygon of side n, sum of all the interior angles = (n – 2) *180^o

Quick Tips for Coordinate Geometry section:

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)]

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