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.
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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
- A right angle is an angle of 90°
- An angle less than 90° is called an acute angle.
- An angle greater than 90° but less than 180° is called an obtuse angle.
- An angle of 180° is a straight line.
- 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.
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