Basic constructions

Hello students, in this session we are going to learn about the Basic Constructions in geometry. In mathematics the geometry help plays a very important role because in this we teaches how to make shapes, figures, lines and angles. A construction in geometry is the skill by which we can able to draw the many types of figures. All constructions will be completed only by the use of ruler and compass as instruments. In every construction we are expected to write down the essential steps of constructions.
Note: - Never draw freehand when doing constructions. (Also refer algebra equation solver to improve your skills)
The list of the basic constructions in geometry is: -
-Copy a line segment.
-Copy an angle.
-Bisect a line segment.
-bisect an angle.
-Construct parallel lines, perpendicular lines and many more.
Mainly we include line, circle and triangle constructions in the geometry constructions.
Line construction is the basic construction for the all figures in geometry. By joining the line we can construct the many figures. And for making the straight line we use ruler and pencil.
Circle construction can be form by using the two ways that are diameter and radius. For making the circle we should have either diameter or radius of the circle. The following steps for constructing the circle: -
Step 1: - Set the compass with the ruler according to required radius.
Step 2: - Then set up the compass' end point on the paper.
Step 3: - Now put the end of the pencil on paper.
Step 4: - Revolve the end of pencil in any direction until we meet the starting point.
Triangle is constructed using ruler, compass and protractor.
Grade XII students can learn basic geometry constructions using the above information.

In upcoming posts we will discuss about circles and Probability and Statistics. Visit our website for information on CBSE 11 physics book

Triangle congruence relationships

Hello students, in this answer math problems for free session we are going to read about the triangle congruence relationships. But before discussing it we will discuss Congruent Triangles, it means âˆ†ABC is said to be congruent to ∆DEF only when one of them can be made to superpose on the other (and vice-versa ) so as to cover it exactly. And, we write ∆ABC ≅ âˆ†DEF. The congruence relation for the triangles is:-
-Every triangle is congruent to itself that is âˆ†ABC ≅ âˆ†ABC.
-If ∆ABC ≅ âˆ†DEF then ∆DEF ≅ âˆ†ABC.
-If ∆ABC ≅ âˆ†DEF then ∆DEF ≅ âˆ†PQR, then ∆ABC ≅ âˆ†PQR.
There is some criteria by which we can also show the triangle congruence relationships, the criteria are: -
SAS (side - angle – side): - If two triangles have two sides and the included angle of the one equal to the corresponding sides and the included angle of the other, then the triangle are congruent.
ASA (angle - side – angle): - If two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle, then the two triangles are congruent.
AAS (angle – angle – side): -If two angles and any side of a triangle are equal to the corresponding angles and side of another triangle than the two triangles are congruent.
SSS (side – side – side): - if the three sides of one triangle are equal to the corresponding three sides of another triangle than the two triangles are congruent.
RHS (Right – angle – Hypotenuse – Side): - Two right angled triangles are congruent if one side and the Hypotenuse of the one are respectively equal to the corresponding side and the Hypotenuse of the other.
Above discussion helps Grade XII students to understand Triangle congruence relationships.

In upcoming posts we will discuss about Basic constructions and Types of events. Visit our website for information on CBSE 10th science syllabus

triangle inequality theorem

Hello students, in this session we are going to discuss the triangle inequality theorem. This theorem defines that the sum of any two sides will always be greater than the third side or we can say that the only one side is shorter than the other two sides; meaning is same in both the scenario. The theorem is - If we have x, y and z sides for the any triangle then,
x + y > z,
y + z > x,
x + z > y,
It is to be noted that if any one side of triangle is greater than the other two sides then we cannot construct the triangle. Also the triangle inequality theorem says same. for more information visit here
We can see this by an example:
Is a triangle with the sides 6cm, 7cm and 8cm possible?
Solution: Sum of 2 sides is always greater than the third side.
Then according to theorem
x + y > z,
y + z > x,
x + z > y,
6 + 7 > 8,
7 + 8 > 6,
8 + 6 > 7,
We can see that in above example the sum of 2 sides is greater than the third side. So, the triangle is possible.
Is a triangle with 3 cm, 4 cm and 13 cm possible?
Solution:
3 + 13 > 4
4 + 13 > 3
3 + 4 >12
In above example the last inequality is false, so, the triangle is not possible
Note:
 i) If in two triangles the two sides are congruent, then the triangle that have larger third side will keep larger included angle.
ii) If in two triangles the two sides are congruent, then the triangle that have larger included angle will keep a larger side.
Central Board of Secondary Education Grade XII students can practice by reading this discussion.

In upcoming posts we will discuss about Triangle congruence relationships and Permutations and combinations. Visit our website for information on Circumference Formula of a Circle

parallel lines cut by a transversal

Hi Friends! In this online tutors homework help session we will talk about parallel lines cut by a transversal. Let us consider two lines which are at equidistance from each other at any point of observation. It means that when we draw a perpendicular at any point from one line to another, we observe that all perpendiculars are of equal length. In this unit we will discuss about the topic parallel lines cut by a transversal. If two parallel lines are intersected by a line which intersect both the lines, then it is called a transversal. To understand parallel lines cut by a transversal definition, we say that a transversal is the line which meets two parallel lines at some point. So a transversal has a point of intersection on both the parallel lines.
Let us consider two parallel lines say ‘l’ and ‘m’. Let ‘n’ be the transversal drawn on the two parallel lines. Now the following   conditions are satisfied:
1.      Since we have l|| m, then the corresponding angles formed by the transversal are equal. These types of angles are four in pairs.
2.      The pair of interior opposite angles is supplementary. These angles are 2 in pairs.
3.      The pair of interior alternate angles so formed is also equal. These angles are 2 in pair.
4.      Also exterior alternate angles are equal. They are also 2 in numbers.
Now keeping these qualities in mind, if one of the angles among all the angles is known, we can find rest of the angles so formed. To find these angles we may use the property of vertical opposite angles are equal, corresponding angles of the two parallel lines, cut by a transversal are equal, Linear pair and the property of alternate angles.
Besides this we come across the problems where we are given some of the measures of the angles (also see Complementary Angles Definition) and we need to find if the two lines which are cut by the transversal are equal or not. This discussion will help students of grade XII to understand the concept of parallel lines cut by a transversal.

In upcoming posts we will discuss about triangle inequality theorem and Measures of central tendency. Visit our website for information on biology syllabus for class 10 ICSE

Angles of triangles and polygons

We say that a polygon is a closed figure with three or more sides. If we talk about What is the Area of a Triangle, we say that a triangle is a closed figure or we call it a polygon with three sides. Here we observe that a triangle has three sides and so it has three angles. Also we must remember that the sum of Angles of triangles is 180 degrees. Thus if we have all the three angles equal, then each angle of the triangle = 180/3 = 60 degrees. Also try area of equilateral triangle calculator to sharpen your skills.
 Now we will look at other polygons. Let us take a three sided figure, say a quadrilateral. We know that a quadrilateral is a four sided figure and it has 4 angles. We also must remember that a quadrilateral is formed by joining 2 triangles. Now if the sum of angles of a triangle is 180 degrees so we say that the sum of angles of a quadrilateral is 180 + 180 = 360 degrees. Thus all the quadrilaterals have an angle sum of 360 degrees.
 Now a polygon can be any figure with 3 or more line segments and when we need to find the Angles of polygons, we must always remember the following formulas:
If the regular polygon is of sides ‘n’, then we sum of have:
1)     Each exterior angle = 360⁰ / n.
2)     Sum of all exterior angles of any polygon = 360⁰.
3)     Also we have each interior angle = 180⁰ - (each exterior angle).
In case of complex polygon of n sides, we have the sum of all exterior angles = 4 right angles = 4 * 90 degrees = 360 degrees.
Also sum of all interior angles = (2*n -4) right angles.
These formulas will help us to find the angles of all the polygons.

In upcoming posts we will discuss about parallel lines cut by a transversal and Conditional probability. Visit our website for information on syllabus of economics for ICSE class 12

Properties of quadrilaterals

In this unit we are going to learn about the Properties of quadrilaterals. The quadrilaterals are the four sided closed figures which are formed by joining the four line segments. All squares, rectangles, parallelograms, trapezium, kite and even all irregular four side figures are called quadrilaterals. A regular quadrilateral is called a square. Here we are going to learn about Define quadrilaterals properties.
We first look at a square: It is a four sided figure with all the sides equal. All the angles of the square are 90 degrees. So we say that it has opposite sides parallel and equal.
Rectangle: A rectangle is a quadrilateral with its opposite sides equal and parallel. Here we have to remember that all the angles are 90 degrees as in square, but all sides are not equal. So we can say it is a square is a special rectangle with its length and breadth as equal.
Parallelogram: A quadrilateral is a four sided figure with its opposite sides parallel and equal. All squares and rectangles are parallelogram, but it is not necessary for all the parallelograms to have its angles as 90 degree, so we conclude that all parallelograms are not necessary a square or the rectangles.
In case of trapezium, we have one pair of opposite sides as parallel, but the pair of parallel lines is not equal. So we come to the conclusion that another pair of opposite lines formed in the trapezium is neither parallel nor equal.
 If we look at a kite, it also has 4 sides so it is called a quadrilateral (for more). In kite we have a pair of adjacent sides equal instead of the pair of opposite sides.
Rhombus is a figure with all four sides equal. It is a tilted form of the square. It has the pair of opposite angles equal but not equal to 90 degrees. In rhombus, we have the diagonals are perpendicular bisector. This discussion will help students of grade XII to understand the Properties of quadrilaterals.

In upcoming posts we will discuss about Angles of triangles and polygons and Mean. Visit our website for information ICSE board syllabus for class 12 math