Geometry Problems with Solutions PDF

2

Geometry Problems with Solutions PDF

INTRODUCTION

Line: A line has length. It has neither width nor thickness. It can be extended indefinitely in both directions.

Ray: A line with one end point is called a ray. The end point is called the origin.

Line segment: A line with two end points is called a segment.

Parallel lines: Two lines, which lie in a plane and do not intersect, are called parallel lines. The distance between two parallel lines is constant.

Perpendicular lines: Two lines, which lie in a plane and intersect each other at right angles are called perpendicular lines.

PROPERTIES

  • Three or more points are said to be collinear if they lie on a line, otherwise they are said to be non-collinear.
  • Two or more lines are said to be coplanar if they lie in the same plane, otherwise they are said to be non-coplanar.
  • A line, which intersects two or more given coplanar lines in distinct points, is called a transversal of the given lines.
  • A line which is perpendicular to a line segment, i.e. intersect at 900 and passes through the midpoint of die segment is called the perpendicular bisector of the segment
  • Every point on the perpendicular bisector of a segment is equidistant from the two endpoints of the segment.
  • If two lines are perpendicular to the same line, they are parallel to each other.
  • Lines which are parallel to the same line are parallel to each other.

Download RRB JE Study Material

Angles: An angle is the union of two non-collinear rays with a common origin. The common origin is called the vertex and the two rays are the sides of the angle.

Congruent angles: Two angles are said to be congruent, denoted by if it divides the interior of the angle into two angles of equal measure.

TYPES OF ANGLE

  1. A right angle is an angle of 90°
  2. An angle less than 90° is called an acute angle.
  3. An angle greater than 90° but less than 180° is called an obtuse angle.
  4. An angle of 180° is a straight line.
  5. An angle greater than 180° but less than 360° is called a reflex angle

PAIRS OF ANGLES

Adjacent angles: Two angles are called adjacent angles if they have a Common side and their interiors are disjoint

Linear Pair: Two angles are said to form a linear pair if they have common side and their other two sides are opposite rays. The sum of the measures of the angles is 180°.

Complementary angles: Two angles whose sum is 90°, are complementary, each one is the complement of the other.

Supplementary angles: Two angles whose sum is 180° are supplementary, each one is the supplement of the other.

Vertically Opposite angles: Two angles are called vertically opposite angles if their sides form two pairs of opposite rays. Vertically opposite angles are congruent

Alternate angles: In the above figure, 3 and 3, 2 and 8 are Alternative angles.

When two lines are intersected by a transverse, they form two pairs of alternate angles.

The pairs of alternate angles thus formed are congruent, i.e. ∠3- ∠3 and ∠2 = ∠8

Interior angles:  When two lines are intersected by a transverse, they form two pairs of interior angles. The pairs of interior angles thus formed are supplementary. i.e. ∠2+∠5-∠3 + ∠8 = 180°

Example 1:

In figure given below, lines PQ and RS intersect each other at point O. If POR; ROQ = 5:7, find all the angles.

Solution:

POR+ROQ = 180° (Linear pair of angles)

But POR: ROQ = 5:7 (Given)

Now, POS = ROQ = 105° (Vertically opposite angles)

and SOQ=POR = 75° (Vertically opposite angles)

PROPORTIONALITY THEOREM

The ratio of intercepts made by three parallel lines on a transversal is equal to the corresponding intercepts made on any other transversal PR/RT=QS/SU

TRIANGLES
The plane figure bounded by the union of three lines, which Join three non-collinear points. Is called a triangle. A triangle la denoted by the symbol ∆.
The three non-collinear points, are called the vertices of the triangle.
In ΔABC, A, B and C are the vertices of the triangle; AB, BC, CA are the three sides, and ∠A, ∠B, ∠C are the three angles.

CLASSIFICATION OF TRIANGLES

Based on sides:
Scalene triangle: A triangle in which none of the three sides is equal is called a scalene triangle.
Isosceles triangle: A triangle in which at least two sides are equal is called an isosceles triangle.
Equilateral triangle: A triangle in which all the three sides are equal is called an equilateral triangle. In an equilateral triangle, all the angles are congruent and equal to 60°.

Based on angles:
Right triangle: If any of a triangle is a right angle i.e., 90° then the triangle is a right-angled triangle.
Acute triangle: If all the three angles of a triangles are acute, i.e., less than 90°, then die triangle is an acute angled triangle.
Obtuse triangle: If any one angle of a triangle is obtuse, i.e., greater than 90°, then the triangle is an obtuse-angled triangle.

SOME BASIC DEFINITIONS

Altitude (height) of a triangle: The perpendicular drawn from the vertex of a triangle to the opposite side is called an altitude of the triangle.

Median of a triangle: The line drawn from a vertex of a triangle to the opposite side such that it bisects the side, is called the median of the triangle. A median bisects the area of the triangle.

Orthocentre: The point of intersection of the three altitudesof a triangle is called the orthocentre. The angle made by any side at the orthocentre = 180°- the opposite angle to the side.

Centroid: The point of intersection of the three medians of a triangle is called the centroid. The centroid divides each median in the ratio 2:1.

Circumcentre: The point of intersection of the perpendicular bisectors of the sides of a triangle is called the circumcentre.

Incentre: The point of intersection of the angle bisectors of a triangle is called the incentre.

Angle bisector divides the opposite sides in the ratio of remaining sides Example: BD/DC=AB/AC=c/b Incentre divides the angle bisectors in the ratio (b+ c):a, (c + a):band(a + b):c

Example: Of the triangles with sides 11, 5, 9 or with sides 6, 10, 8; which is a right triangle?

Solution:

(Longest side)2= 112 – 121;
52+92=25 + 81 = 106
∴ 112≠ 52 + 92
So, it is not a right triangle
Again,(longest side)2 = (10)2 = 100;
62 + 82 = 36 + 64= 100
102 = 62 + 82
∴It is a right triangle.

Example 15:

In figure, ∠DBA = 132° and ∠EAC = 120°. Show that AB>AC.

Solution:

As DBC is a straight line,

132°+∠ABC = 180°

= ∠ABC =180°-132°=48°

For ∆ ABC,

∠EAC is an exterior angle

120° = ∠ABC+∠BCA

(ext. ∠= sum of two opp. interior ∠ s)

⇒120° = 48°+ ∠ BCA

⇒∠BCA=120°-48°=72°

Thus, we find that ∠BCA>∠ABC

⇒ AB > AC (side opposite to greater angle is greater)

POLYGON

A plane figure formed by three or more non-collinear points joined by line segments is called a polygon.

  • A polygon with 3 sides is called a triangle.
  • A polygon with 4 sides is called a quadrilateral.
  • A polygon with 5 sides is called a pentagon.
  • A polygon with 6 sides is called a hexagon.
  • A polygon with 7 sides is called a heptagon.
  • A polygon with 8 sides is called an octagon.
  • A polygon with 9 sides is called a nonagon.
  • A polygon with 10 sides is called a decagon.

Regular polygon:

A polygon in which all its sides and angles are equal, is called a regular polygon. Sum of all interior angles of a regular polygon of side n is given by (2n – 4) 90°. Hence, angle of a regular polygon = ((2n-4)90°)/n Sum of an interior angle and its adjacent exterior angle is 180°
Sum of all exterior angles of a polygon taken in order is 360°.

Example 21:

The sum of the measures of the angles of regular polygon is 2160°. How many sides    does it have?

Solution:

Sum of all angles = 90° (2n – 4)

⇒ 2160 = 90 (2n – 4)

2n = 24 + 4

∴ n = 14

Hence the polygon has 14 sides.

CIRCLE:

The collection of all points in a plane, which are at a fixed distance from a fixed point in the plane, is called a circle. The fixed point is called the centre of the circle and the fixed distance is called the radius r.

Example: The point A divides the join the points (-5, 1) and (3, 5) in the ratio k: l and coordinates of     points B and C are (1, 5) and (7, -2) respectively. If the area of ∆ ABC be 2 units,    then find the value (s) of k.

To Get More Examples and Exercise Download PDF…

Geometry Study Material PDF Download
Download Exercise Questions with Answer Key Pdf

Download Quantitative Aptitude Study Material 

RRB  WhatsAPP Group – Click Here

Telegram Channel   Click Here

Join Us on FB   Examsdaily

Join Us on  Twitter – Examsdaily

2 COMMENTS

LEAVE A REPLY

Please enter your comment!
Please enter your name here