Introduction:
We will have a look at some of the attributes of important shapes in this article.
Triangle:
A triangle is a closed figure with three sides. Any three points that don’t lie on a straight line can form a triangle. A triangle is defined by its sides and the angles opposite to these sides. The points at which these sides meet are called the vertices. An angle is formed at each of these vertices.
Examples of Triangles in Real Life:
The triangle in the figure is named ∆ABC.
The vertices are A, B and C.
Sides are the segments AB, BC and AC.
Depending on the characteristics of the three sides (and the angles they enclose), we can classify triangles.
Classification
The classification of triangles is based on certain criteria:
- Length of sides:
Type | Characteristic | Image |
Equilateral | All sides of the triangle are equal. | |
Isosceles | Any two sides of the triangle are equal. | |
Scalene | All sides of the triangle are unequal. |
- Value of angles:
The sum of all angles of a triangle is 180°. Recall that an angle that measures more than 90° is called an obtuse angle and one that measures less than 90° is an obtuse angle. Also, an angle that measures 90° is a right angle.
Type | Characteristic | Image |
Right-angled | One angle of the triangle has a measure of 90°. | |
Acute-angled | All angles of the triangle measure | |
Obtuse-angled | All sides of the triangle are unequal. |
TIP!
- Angles that are opposite to equal sides have the same measure.
- In an equilateral triangle, all angles have the same measure.
- In an isosceles triangle, angles opposite to equal sides are equal in measure.
- In a right angled triangle, the side opposite to 90° is called its hypotenuse.
THINK!
What could be the measure of an angle in an equilateral triangle?
As the sum of angles of a triangle is 180°, each of the three angles will measure $\frac{180}{3}$ i.e, 60°.
Quadrilateral
A closed figure with 4 sides is called a quadrilateral.
Apart from vertices and sides, a building block of a quadrilateral is also its diagonal.
A diagonal is any segment that joins two non-adjacent vertices of a closed figure.
An important characteristic to note about a quadrilateral is that the sum of its interior angles is 360°.
Any quadrilateral can be broken down into two triangles with a common base form a quadrilateral.
There are a number of types of quadrilaterals that are important in Mathematics. Let us look at them one by one.
- Parallelogram
By definition, it is a quadrilateral whose opposite sides are parallel.
Some of its other characteristics include:
- The opposite sides have equal length.
Here, AB=CD; AD=BC - The opposite angles are equal in measure.
∠A=∠C; ∠B=∠D - The adjacent angles are supplementary (their sum is 180°)
∠A+∠B=∠C+∠D=180°
- Rectangle
- A rectangle is a special type of a parallelogram whose angles are all measured 90°.
So, here, ∠A=∠B=∠C=∠D=90° - For a rectangle, the diagonals are equal in length.
Here, AC=BD. - The diagonals also divide each other into equal parts.
EXAMPLE:
Sheet of paper
- Rhombus
- A rhombus is a special parallelogram with all sides having the same measure.
Here, AB=BC=CD=AD. - The diagonals of a rhombus meet at an angle of 90°.
EXAMPLE:
- Square
A square is a quadrilateral that combines the characteristics of a rectangle and a rhombus.
- All sides of a square are equal in measure.
- All angles are measured 90°.
- The diagonals are equal in length.
- The diagonals of a rhombus meet at an angle of 90°.
EXAMPLE:
- Trapezium
- A trapezium is a quadrilateral with one pair of opposite sides being parallel.
- In an isosceles trapezium, the non-parallel sides have equal length.
- Here, AD||BC;AB=CD
EXAMPLE:
- Kite
- A kite is a quadrilateral which can be broken down into two isosceles triangles.
- Each side is equal in length to exactly one adjacent side.
- In the given figure, AB=AD and BC=BD.
- Thus, ∆ABD and ∆ACD are isosceles triangles.
- As you might have guessed, the name of this shape comes from most common types of kites.
- The diagonals meet at right angles.
Polygons
Any closed figure with n number of sides such that n≥3 is a polygon. Thus, triangles and quadrilaterals are polygons as well.
A polygon with all sides having equal length is called a regular polygon.
Some higher polygons are:
- Pentagon
A pentagon has 5 sides. | Diamond |
- Hexagon
A hexagon has 6 sides. | Beehive |
In a regular hexagon, the length of diagonal is twice the length of sides.
- Septagon
A septagon has 7 sides. It is also called a heptagon.
- Octagon
An octagon has 8 sides.
- Nonagon
A nonagon has 9 sides
- Decagon
A decagon has 10 sides.
Angles of a Polygon
- Interior angles
- For any polygon, the angles measured between any two adjacent edges(or sides) within the polygon is called its interior angle.
- For an n-sided polygon, the sum of interior angles is (n-2)×180°.
- Thus the sum of interior angles of an octagon, which has 8 sides is
= (8-2)×180°
=6×180°
=1080° - In a regular polygon, each interior angle has an equal measure. Thus, in a regular octagon, each angle measures
$\frac{1080}{8}$=135°
- Exterior angles
- The angles between the consecutive edges of a polygon, measured outside the polygon are the exterior angles.
- The sum of exterior angles of a polygon is always 360°.
- The exterior angle of a polygon forms a linear pair with the interior angle and can thus be easily calculated if the value of the interior angle is known.
Regular Polygon | Value of n | Sum of interior angles | Measure of interior angle | Measure of exterior angle |
Pentagon | n=5 | (n-2)×180° =(5-2)×180° =3×180° =540° | $\frac{540}{5}$=108° | 180-108 =72° |
Hexagon | n=6 | (n-2)×180° =(6-2)×180° =4×180° =720° | $\frac{720}{6}$=120° | 180-120 =60° |
Septagon | n=7 | (n-2)×180° =(7-2)×180° =5×180° =900° | $\frac{900}{7}$=128.57° | 180-128.57 =51.43° |
Concave and convex polygons:
- If each interior angle of a polygon is less than 180°, the polygon is a convex polygon.
- If the measure of any interior angle is greater than 180°, the polygon is a concave polygon.
Circle
- A circle is a continuous curved object such that every point lying on it is at a fixed distance from a fixed point.
- The fixed point is called the centre of a circle.
- The fixed distance is called the radius.
- The radius of a circle may be defined as a segment joining the centre to any point lying on it.
- Consider the circle with centre O and a point P lying on the circle.
- OP is the radius of this circle.
- The distance of the centre O from any other point, for example Q, that
lies on the circle will be the same as the distance OP.
Important lines/segments in a circle
- Chord
- A segment that touches the circle at exactly two points is called a chord.
- A circle can have infinitely many chords, of varying lengths.
- These chords may pass through any point. in the circle.
- In the adjoining figure, PQ is a chord of a circle that touches it at Exactly two points: P and Q.
- Diameter
- A diameter is a special chord of a circle that must pass through the centre.
- In the adjoining figure, PR is a diameter of the circle.
- Since OP is the radius, we can say that the length of the diameter is twice that of the radius.
- The centre of the circle is also the midpoint of its diameter.
- There can be infinitely many diameters in a circle, each of them having a fixed length.
- Secant
- A secant is a line that passes through a circle at exactly two points.
- There can be infinitely many secants passing through different points of a circle.
- In the adjoining figure, line m is a secant to the circle passing through the points A and B.
- Tangent
- A Tangent is a line drawn from a point outside a circle that touches it at exactly one point.
- The point at which the tangent touches the circle is called its point of contact.
- A tangent is perpendicular to the radius drawn at the point of contact.
- From any external point, exactly two tangents may be drawn to a circle.
- Tangents drawn from the same point to a given circle are equal in length.
- In the adjoining figure, lines s and r are tangents to the circle drawn from the point T.
- The lengths of segments TQ and TP are equal.
Important building blocks of a circle
- Segment
- Any chord of a circle divides it into two segments.
- In the adjoining figure, chord AB divides the circle into two segments: APB and AQB.
- Semicircle
- If the chord dividing the circle is its diameter, the two segments formed are semicircles.
- Here, the diameter AB divides the circle into two semicircles, AQB and APB.
- Sector
- The reqion enclosed within two radii of a circle is called its sector.
- In the given circle, radii OA and OB include two sectors: OAPB and OAQB.
- OAPB is the minor sector and OAQB is the major sector
- Quadrant
- If the angle enclosed within the two radii measures 90°, then the minor sector formed is called a quadrant of the circle.
- In the given circle, AOQP is a quadrant.
- A circle may be divided into 4 quadrants.
Line of Symmetry:
- A line of symmetry divides a shape into two parts such that each part is a mirror image of the other.
- The division should be such that when a shape is folded about the line, the two parts should perfectly match up to each other.
- Lines of symmetry of some common images can be found in the table below:
Shape | Figure | Lines of symmetry |
Scalene Triangle | Zero | |
Isosceles Triangle | One | |
Equilateral Triangle | Three | |
Rectangle | Two | |
Square | Four | |
Isosceles Trapezium | One | |
Kite | One | |
Parallelogram | Zero | |
Regular Pentagon | Five | |
Regular Hexagon | Six | |
Circle | Infinite | |
Semicircle | One |
TIP!
- A line of symmetry passing through a side divides it into two equal parts.
- An n-sided regular polygon has n lines of symmetry
- Every diameter of a circle is its line of symmetry.
- All lines of symmetry of a particular shape pass through a single centre, i.e, they are concurrent.
AREAS AND PERIMETERS:
- The region included within a shape is called its area.
- The border of a shape is called its perimeter.
- For circular figures, the perimeter is also referred to as the circumference.
Areas and perimeters of triangles and quadrilaterals:
Shape | Figure | Perimeter | Area |
Triangle | Sum of sides =AB+BC+AC | $\frac{1}{2}$×base×height =$\frac{1}{2}$×AD×BC | |
Equilateral triangle | 3×side =3×AB | $\frac{√3}{4}$ x (side)^{2} =$\frac{√3}{4}$ x (AB)^{2} | |
Parallelogram | 2×(AB+BC) | AB×BC | |
Square | 4×side =4×AB | (side)^{2} =(AB)^{2} | |
Trapezium | Sum of sides =AB+BC+CD+AD | $\frac{1}{2}$×(sum of parallel sides)×height =$\frac{1}{2}$ s AD+BC×AE |
The Number pi (π)
- The ratio of the diameter of any circle and its circumference is always constant.
- The value of this constant is approximate 3.14 or $\frac{22}{7}$.
- This constant is called ‘pi’ and is denoted by the symbol π
Arc of a circle
- An arc is a part of the circumference of a circle enclosed between two radii.
- In the given circle, APB and AQB are two arcs enclosed within the Radii OA and OB
- APB is the minor arc of the circle and AQB is the major arc.
Areas and circumferences of circular shapes:
Shape | Figure | Circumference | Area |
Circle | Radius=r, Diameter=d | 2πr OR πd | r^{2} |
Semicircle | πr | $\frac{πr^2}{2}$ | |
Quadrant | $\frac{πr}{2}$ | $\frac{πr^2}{4}$ |
Problems:
- The quadrilateral in the given figure is a rectangle.
If ∠BAC=30°, find the measure of ∠ACD.
Solution:
In ∆ABC,
∠ABC=90°… [Angle of a rectangle]
∠ABC+∠BAC+∠ACB=180°… [Angles of a triangle]
90+30+∠ACB=180
120+∠ACB=180
∠ACB=180-120
∠ACB=60°.
Now, ∠ACD+∠ACB=∠BCD
∠BCD=90° [Angle of a rectangle].
∠ACD+∠ACB=90
∠ACD+60=90
∠ACD=30°.
Answer:
The measure of ∠ACD is 30°.
- In the given quadrilateral, find the measure of ∠C if ∠A=40°, ∠B=120°, ∠D=90°.
Solution:
∠A+∠B+∠C+∠D=360° … [Angles of a quadrilateral]
40+120+∠C+90=360
250+∠C=360
∠C=360-250
∠C=110°.
Answer:
The measure of ∠C is 110°.
- In the given parallelogram, if ∠C=150°, find the missing angles.
Solution:
In Parallelogram ABCD,
∠A=∠C… [Opposite angles of a parallelogram]
∠A=150°
Now,
∠A+∠B=180° … [Adjacent angles of a parallelogram]
150+∠B=180
∠B=180-150
∠B=30°
Also,
∠D=∠B…[Opposite angles of a parallelogram]
∠D=30°
Answer:
The measures of the remaining angles are 30°, 180°, 30°.
- In the given isosceles right-angled triangle, find the measure of ∠ACD.
Solution:
In ∆ABC,
∠ABC=90° … [Right Angle]
AB=BC … [Sides of isosceles triangle]
∠ACB=∠BAC … [Opposite angles of equal sides]
Thus,
∠ACB=∠BAC=45°.
Now,
∠ACB+∠ACD=180° … [Angles in a linear pair]
45+∠ACD=180
∠ACD=180-45
∠ACD=135°
Answer:
The measure of ∠ACD is 135°.
- Calculate the measure of ∠AOD if the given quadrilateral is a rectangle and ∠ADO=75°.
Solution:
In ∆AOD,
AO=OD … [Diagonals of a rectangle are equal and bisect each other]
∠ADO=∠DAO=75° … [Angles opposite to equal sides]
Now,
∠ADO+∠DAO+∠AOD=180° … [Angles of a triangle]
75+75+∠AOD=180
150+∠AOD=180
∠AOD=180-150
∠AOD=30°
Answer:
The measure of ∠ACD is 135°.
- Find the measure of ∠AOB in the regular hexagon.
Solution:
In the regular hexagon ABCDEF,
Diagonals AD=2 Side AB
Thus, AO=OB=AB
∆AOB is an equilateral triangle.
∠AOB=60° … [Angle of an equilateral triangle]
Answer:
The measure of ∠AOB is 60°.
- Given a kite, if the measure of ∠OAD=18°, find ∠ABD.
Solution:
In ∆AOD,
∠AOD=90° … [Diagonals of a kite are perpendicular]
∠AOD+∠ADO+∠DAO=180° … [Angles of a triangle]
90+18+∠ADO=180
108+∠ADO=180
∠ADO=180-108
∠ADO=72°
∠ADB=72° … [Angles in the same line]
∆ABD is an isosceles triangle.
∠ABD=∠ADB … [Angles opposite to equal sides]
∠ABD=72°
Answer:
The measure of ∠ABD is 72°.
- Given ∆PQR is an isosceles triangle with PQ=PR.
If ∠Q=70°, find the measure of ∠P.
Solution:
Here, PQ=PR.
∴∠R=∠Q=30°
Now,
∠P+∠Q+∠R=180° … [Angles of a triangle]
∠P+70+70=180
∠P+140=180
∠P=180-140
∠P=40°
Answer:
The measure of ∠P is 40°.
- If the area of a circle is 9π, find the circumference of the corresponding semicircle.
Solution:
Area of a circle=πr^{2}
r^{2}=9π
r^{2}=9π
r^{2}=9
r=√9
r=3
Circumference of semicircle
=πr
=π×3
=3π
Circumference of semicircle=3π
Answer:
The circumference of semicircle is 3 units.
- Given isosceles trapezium ABCD, find the area of rectangle ADFE, If the area of the trapezium is 72.
Solution:
Area of trapezium=$\frac{1}{2}$×(AD+BC)×AE
$\frac{1}{2}$ x (8+12)×AE=72
$\frac{1}{2}$×20×AE=72
10×AE=72
AE=$\frac{72}{10}$
AE=7.2
Area of rectangle
=AD×AE
=8×7.2
=57.6
Answer:
The area of the rectangle is 57.6 sq units.
- Given a regular hexagon ABCDEF that has a perimeter 144, find the length of diagonal AD.
Solution:
In a regular hexagon, all sides are equal in length.
So,
Perimeter=6×side
144=6×BC
BC=$\frac{144}{6}$
BC=24
Also,
diagonal=2×side
AD=2×BC
AD=2×24
AD=48.
Answer:The length of diagonal is 48 units.
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