Class 9 Mathematics | Punjab Curriculum and Textbook Board Syllabus 2025
Similar figures have the same shape but not necessarily the same size.
Two polygons are similar if:
Note: Proportionality of sides means one side is \( k \) times its corresponding side.
A three or more than three-sided closed figure is called a polygon.
If two angles in one triangle are congruent to two corresponding angles in another triangle, the third angle in each triangle must also be congruent. Since the angles are the same, the triangles are similar. The similarity symbol is \( \sim \).
Example: In the correspondence of triangles \( ABC \) and \( DEF \), if:
\( m\angle A = m\angle D \)
\( m\angle B = m\angle E \)
\( m\angle C = m\angle F \)
then triangle \( \Delta ABC \sim \Delta DEF \).
The ratio of areas of two similar figures is equal to the square of the ratio of their corresponding lengths.
Where:
Two solids are said to be similar if they have the same shape but possibly different sizes.
Example: Two cylinders are similar if:
where \( r_{1} \), \( r_{2} \) are the radii and \( h_{1} \), \( h_{2} \) are the heights of the two cylinders.
For two similar solids, the ratio of their volumes is equal to the cube of the ratio of their corresponding lengths:
For solids made of the same material, mass is directly proportional to volume:
So:
where \( w_{1} \) and \( w_{2} \) are the masses of the similar solids.
A regular polygon has all sides and all angles equal. Some of the common regular polygons are equilateral triangles, squares, regular pentagons, regular hexagons, etc.
The formula for sum of interior angles of an \( n \)-sided polygon is:
For a regular \( n \)-sided polygon:
Example: For a regular hexagon (\( n = 6 \)):
\[ \text{Each interior angle} = \frac{(6 - 2) \times 180^\circ}{6} \]
\[ = \frac{720^\circ}{6} \]
\[ = 120^\circ \]
The sum of all exterior angles of any polygon is always \( 360^\circ \), regardless of the number of sides.
The exterior angle of a regular \( n \)-sided polygon is:
Note: The interior and exterior angles are supplementary at a vertex. That is:
The total number of diagonals in a regular polygon with \( n \)-sided is:
A regular \( n \)-sided polygon has rotational symmetry and reflexive (line) symmetry, both of order \( n \).
Example: A regular hexagon has six lines of symmetry and rotational symmetry of order 6. It can be rotated by \( \frac{360^\circ}{n} \) and still look the same.
A triangle is a polygon with three sides and three angles.
Triangles come in various types based on side length and angle measure:
The measure of an exterior angle in a triangle is equal to the sum of the measures of two opposite interior angles.
In \( \Delta ABC \):
\[ m\angle A + m\angle B = m\angle BCD \]
\[ x + y = w \]
A parallelogram is a quadrilateral whose opposite sides are parallel and equal in length and opposite angles are equal. Its adjacent angles are supplementary.
The diagonals of a parallelogram bisect each other (they cross each other at the midpoint). They are not equal in length.
A tessellation is a pattern of shapes that fit together perfectly without any gaps or overlaps to completely cover a flat surface.
Only three regular polygons can tessellate the plane on their own:
Regular pentagons and other polygons with angles that don't add up to \( 360^\circ \) at each vertex cannot form gap-free patterns, so tessellation is not possible.
Polygons are used in:
| Regular Tessellation | Irregular Tessellation |
|---|---|
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A pattern of shapes that fit together perfectly without any gaps or overlaps to completely cover a flat surface. Uses same regular polygons only. Tessellations using:
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A pattern made using different shapes: regular polygons, irregular polygons, or both. Uses different or irregular polygons, or a mix of both. Tessellations using:
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