Co-ordinate Geometry Distance and Section Formula
- Compound Interest as a Repeated Simple Interest Computation with a Growing Principal
- Use of Compound Interest in Computing Amount Over a Period of 2 Or 3-years
- Use of Formula
- Finding CI from the Relation CI = A – P
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Areas of Sector and Segment of a Circle
- Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments
- Tangent Properties - If a Chord and a Tangent Intersect Externally, Then the Product of the Lengths of Segments of the Chord is Equal to the Square of the Length of the Tangent from the Point of Contact to the Point of Intersection
- Tangent to a Circle
- Number of Tangents from a Point on a Circle
- Chord Properties - a Straight Line Drawn from the Center of a Circle to Bisect a Chord Which is Not a Diameter is at Right Angles to the Chord
- Chord Properties - the Perpendicular to a Chord from the Center Bisects the Chord (Without Proof)
- Theorem: Equal chords of a circle are equidistant from the centre.
- Theorem : The Chords of a Circle Which Are Equidistant from the Centre Are Equal.
- Chord Properties - There is One and Only One Circle that Passes Through Three Given Points Not in a Straight Line
- Arc and Chord Properties - the Angle that an Arc of a Circle Subtends at the Center is Double that Which It Subtends at Any Point on the Remaining Part of the Circle
- Theorem: Angles in the Same Segment of a Circle Are Equal.
- Arc and Chord Properties - Angle in a Semi-circle is a Right Angle
- Arc and Chord Properties - If Two Arcs Subtend Equal Angles at the Center, They Are Equal, and Its Converse
- Arc and Chord Properties - If Two Chords Are Equal, They Cut off Equal Arcs, and Its Converse (Without Proof)
- Arc and Chord Properties - If Two Chords Intersect Internally Or Externally Then the Product of the Lengths of the Segments Are Equal
- Cyclic Properties
- Tangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers
Shares and Dividends
- Circumscribing and Inscribing a Circle on a Regular Hexagon
- Circumscribing and Inscribing a Circle on a Triangle
- Construction of Tangents to a Circle
- Circumference of a Circle
- Circumscribing and Inscribing Circle on a Quadrilateral
Ratio and Proportion
Gst (Goods and Services Tax)
- Sales Tax, Value Added Tax, and Good and Services Tax
- Computation of Tax
- Concept of Discount
- List Price
- Concepts of Cost Price, Selling Price, Total Cost Price, and Profit and Loss, Discount, Overhead Expenses and GST
- Basic/Cost Price Including Inverse Cases.
- Selling Price
- Goods and Service Tax (Gst)
- Gst Tax Calculation
- Gst Tax Calculation
- Input Tax Credit (Itc)
- Linear Inequations in One Variable
- Solving Algebraically and Writing the Solution in Set Notation Form
- Representation of Solution on the Number Line
- Arithmetic Progression - Finding Their General Term
- Sum of First ‘n’ Terms of an Arithmetic Progressions
- Simple Applications of Arithmetic Progression
- Median of Grouped Data
- Graphical Representation of Data as Histograms
- Ogives (Cumulative Frequency Graphs)
- Concepts of Statistics
- Graphical Representation of Data as Histograms
- Graphical Representation of Ogives
- Finding the Mode from the Histogram
- Finding the Mode from the Upper Quartile
- Finding the Mode from the Lower Quartile
- Finding the Median, upper quartile, lower quartile from the Ogive
- Calculation of Lower, Upper, Inter, Semi-Inter Quartile Range
- Concept of Median
- Mean of Grouped Data
- Mean of Ungrouped Data
- Median of Ungrouped Data
- Mode of Ungrouped Data
- Mode of Grouped Data
- Mean of Continuous Distribution
- Geometric Progression - Finding Their General Term.
- Geometric Progression - Finding Sum of Their First ‘N’ Terms
- Simple Applications - Geometric Progression
Co-ordinate Geometry Equation of a Line
- Slope of a Line
- Concept of Slope
- Equation of a Line
- Various Forms of Straight Lines
- General Equation of a Line
- Slope – Intercept Form
- Two - Point Form
- Geometric Understanding of ‘m’ as Slope Or Gradient Or tanθ Where θ Is the Angle the Line Makes with the Positive Direction of the x Axis
- Geometric Understanding of c as the y-intercept Or the Ordinate of the Point Where the Line Intercepts the y Axis Or the Point on the Line Where x=0
- Conditions for Two Lines to Be Parallel Or Perpendicular
- Simple Applications of All Co-ordinate Geometry.
- Figures with one Line of Symmetry
- Figures with two Line of Symmetry
- Figures with more Line of Symmetry
- Line Symmetry: A figure has line symmetry if a line can be drawn dividing the figure into two symmetrical parts. The line is called a line of symmetry.
A figure has line symmetry if a line can be drawn dividing the figure into two symmetrical parts. The line is called a line of symmetry.
A figure can have any number of lines of symmetry passing through it. Some can have one, two, or even multiple lines of symmetry.
1. Figures with one Line of Symmetry:
A figure has line symmetry if there is a line about which the figure may be folded so that the two parts of the figure will coincide.
2. Figures with two Lines of Symmetry:
Example of two lines of symmetry: A rectangle.
- Take a rectangular sheet (like a post-card).
- Fold it once lengthwise so that one-half fits exactly over the other half.
- Open it up now and again fold on its width in the same way.
- After opening it, we get two lines of symmetry of the rectangular sheet.
Construction of figure with two Lines of Symmetry:
We can also learn to construct figures with two lines of symmetry starting from a small part.
1. Let us have a figure as shown alongside.
2. We want to complete it so that we get a figure with two lines of symmetry. Let the two lines of symmetry be L and M.
3. We draw the part as shown to get a figure having line L as a line of symmetry.
4. To complete the figure we need it to be symmetrical about line M also. Draw the remaining part of the figure as shown. This figure has two lines of symmetry i.e. line Land line M.
3. Figures with multiple Lines of Symmetry:
Example of multiple lines of symmetry (two or more)
- Take a square piece of paper. Fold it into half vertically, fold it again into half horizontally. (i.e. you have to fold it twice).
- Now open out the folds and again fold the square into half (for a third time now), but this time along a diagonal.
- Again open it and fold it into half (for the fourth time), but this time along the other diagonal.
- When you will open the paper you will see four imaginary lines and these lines are the lines of symmetry.
Some more images with more than two lines of symmetry:
The equilateral triangle will have three lines of symmetry.
Square will have four lines of symmetry.
The regular pentagon will have five lines of symmetry.
Regular hexagon will have six lines of symmetry.
Shaalaa.com | Figures Having Two Or Multiple Lines Of Symmetry Part-1
Series: Concept of Lines Symmetry
Draw an isosceles ΔABC, where BC = 3.5 cm, the base angles C and B = 75°. Use ruler and compass only. Draw all lines of symmetry of the triangle.
A(2,2) and B(5,5) are the vertices of a figure which is symmetrical about xaxis. Complete the figure and give its geometrical name.
In the following figures, the line of symmetry has been drawn with a dotted line. Identify the corresponding sides and the corresponding angles about the line of symmetry.