Statistics

Statistics

Points to be Remember:-

1. Pictorial representation of the numerical data using picture

     symbols is called pictograph of the data.

2. A bar graph is a pictorial representation of the numerical data by a

    number of bars of uniform  width, erected horizontally or vertically,

    with equal spacing between them.

3. The uniform width of the bars and the uniform gap between them is

    suitably chosen, keeping in view the given information (data) and

    the space available for the diagram.

4. it is not necessary to make the bar graph on a graph paper, but

    from the point of view of convenience and accuracy, one should do

    so.

5. A bar graph should have a little. The chosen scale should also be

    mentioned below the graph.

Surface Areas and Volumes

Surface Areas and Volumes

Points to be Remember:-

1. A cuboid is a figure in space, formed by three pairs of opposite

    congruent rectangular faces, such that whenever two faces meet,

    they meet in a line segment.

2. A cuboid has 6 faces, 12 edges and 8 vertices.

3. The part of the space enclosed by a cuboid is called its interior.

4. A cuboid along with its interior is called a cuboidal region.

5. The length, breadth and height of a cuboid are called its three

     dimensions.

6. A cuboid whose length, breadth and height of a cuboid are called a 

    cube.

7. The magnitude or measure of a space (solid) region is called its

     volume.

8. A cubic centimeters is the volume of the region formed by a cube

    of side ( or edge ) 1 cm.

9. With usual meaning for V, I, b and h

     (i) Surface area of a cuboid = 2 (lb + bh + hl)

     (ii) Surface area of a cube = 6l²

    (iii) Lateral surface area of a cuboid = 2 (l + b)h

    (iv) Volume of a cuboid (V) = l x b x h

    (v) Volume of a cube (V) = l³

10. Standard units of volume

       (i) 1m³ = 1000000cm³ = 100³cm³

       (ii) 1cm³ = 1000mm³ = 10³mm³

      

       (iii) 1cm³ = 1000l = 1kl

       (iv) 1m3 = 1000l = 1kl

Areas to rectangle

Areas to rectangle

Points to be Remember:-

1. Area of a rectangle  =  length  x  breadth

2. Area of a square  =  ( side )²

3. Area of a rectangular path inside ( or outside) a rectangular field        

     = Area of the outer rectangle - Area of the inner rectangle.

4. Area of a cross path = Area of all the rectangles making the paths

    – Area of the common rectangle ( or square).

Circles

Circles

Points to be Remember:-

1. The region bounded by an arc of the circle and the chord of the arc

     is called a segment of the circle.

2. The part of the plane that consists of a diameter, semi-circle and

     interior of the circle that is enclosed by the semi-circle and the

     diameter is called a semi-circle region.

3. Angle in a semi-circle is a right angle.

4. Angles formed in the same segment are equal.

Quadrilaterals

Quadrilaterals

Points to be Remember:-

1. If we take four points say A,  B, C, and D in a plane such that no

    three of them are collinear and the line segments AB, BC, CD and  

    DA intersect at their ends only, then the figure so formed by the

    four line segments is called a quadrilateral.

2. The points A,  B, C, and D are called the vertices of the

    quadrilateral ABCD.

3. The line segments AB, BC, CD and DA are called the sides of the

     quadrilateral ABCD.

4. The angles of the quadrilateral ABCD are ÐA, ÐB, ÐC, and ÐD.

5. The two line segments joining the opposite vertices are called the

    diagonals of the quadrilateral ABCD.

6. In a quadrilateral, two sides that have a common vertex are known

    as adjacent sides.

7. In a quadrilateral, any two sides that do not have a vertex in

    common are called opposite sides.

8. Two angles of a quadrilateral are adjacent angles, if they have a

    side of the quadrilateral as a common arm.

9. Two angles of a quadrilateral that are not adjacent are called

     opposite angles.

10. The interior of the quadrilateral ABCD, together with its boundary

       (i.e. the quadrilateral itself) is called the quadrilateral region

       ABCD.

11. The sum of the angle of a quadrilateral is 360°.

Congruent Triangles

Congruent Triangles

Points to be Remember:-

1. Two figures are congruent, if they have the same shape and size.

2. Two triangles are congruent, if in a matching of their vertices, the

     three sides and the three angles of one triangle are respectively

     equal to the corresponding parts of the other.

3. Two triangles are congruent, if two sides and the included angle of

     the one triangle are respectively equal to the two sides and the

     included angle of the other (SAS congruence condition)

    SSS congruence condition

   Two triangles are congruent, if the three sides of the triangle are

    respectively equal to the three sides of the other. This is called

    SSS congruence condition.

     RHS congruence condition

  Two right triangles are congruent, if the hypotenuse and one side of  

  the one triangle are respectively equal to the hypotenuse and one

  side of the other. This is called RHS congruence condition.

More about Triangles

More about Triangles

Points to be Remember:-

1. If two sides of a triangle are equal, then their opposite angles are

     also equal.

2. If two angles of a triangle are equal, then their opposite sides are

    also equal.

Points to be Remember:-

1. Pythagoras theorem: In a right angled triangle, the square of

    hypotenuse equals the sum of the square of its side

2. In a right angled triangle, the hypotenuse is the longest side.

3. If the square of one side of a triangle is equal to the sum of the

    squares of the remaining two sides, then the angle opposite the

    first side is a right angle.

Points to be Remember:-

1. The perpendicular line segment drawn from any vertex of a triangle 

     into the opposite side is called altitude of the triangle.

2. A triangle has three altitudes. From each vertex one altitude is

    Obtained

3. The altitudes of a triangle are concurrent. The point of

     concurrence of the altitudes of a triangle is called the orthocenter

     of the triangle.

4. The line segment joining any vertex of a triangle to the mid-point of

     its opposite side is known as median of the triangle.

5. A triangle has three vertices and from each vertex one median is

     obtained. Thus a triangle has three medians.

6. The medians of a triangle are concurrent. The point of concurrence

     of the medians of a triangle is called centroid of triangle.

Points to be Remember:-

1. A perpendicular bisector of a side of a triangle is the line that is

    perpendicular to the side and bisects it.

2. A triangle has three sides. Therefore, it has three perpendicular

    bisects.

3. The perpendicular bisects of the sides of a triangle are concurrent.

    The point of concurrence of the perpendicular bisectors of the 

    sides of a triangle is called the circumcentre of the triangle.

4. An angle bisector of a triangle is the line segment which bisects a

     angle of the triangle and has its other end point on the opposite

     side of that angle.

5. A triangle has three angles and each angle has a bisector.

     Therefore a triangle has three angle bisector.

6. The angle bisector of a triangle are concurrent. The point of

     concurrence of the angle bisectors of a triangle is called its

     incentre.