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A triangle is a geometrically closed figure which is bounded by three lines and angles. It is categorized into two types as sides and angles.
We can categorize the triangle into following 3 types on the basis of sides:
We can categorize the triangle into following 3 types on the basis of angle:
A right-angled triangle which is taken into consideration is known as the reference angle. For example, Assuming \(\angle\) EFG as the reference angle. where angle E is known as remaining angle. So,opposite to the right angle is known as hypotenuse and it is denoted by h. The side opposite to the reference angle, the side is called perpendicular which is denoted by p and remaining side are known as the base and it is denoted by b ,
Here,
EF = hypotenuse(h)
EG=Perpendicular (p)
FG= base(b)
In trigonometry, most of the reference angles are represent by verious Greek alphabets. Some of them are given below:
alpha | \(\alpha\) |
beta | \(\beta\) |
gamma | \(\gamma\) |
delta | \(\delta\) |
We normally use these symbols for representing reference angles.
For example,
Here, \(\triangle\) EFG is a right angled triangle, right angled at F.
Let, ∠GEF = θ be the reference angles .
Then, side EG = hypotenuse ( h ) [\(\therefore\)It is opposite to right angle ]
side EF = perpendicular ( p ) [\(\therefore\)It is opposite to reference angle ]
side FG = base ( b ) [\(\therefore\)It is side opposite to remaining angle ]
Pythagoras Theoramem
Perpendicular, Base and Hypotenuse is established by a famous mathematician named Pythagoras. Pythagoras Theorem is a relation developed on the basis of the elements of right angled triangle.
Here, Pythagoras theorem states that the sum of squares of the perpendicular and base of a right-angled triangle is equal to the squares of the hypotenuse . In simple language , the square made on the hypotenuse of a right-angled triangle is always equal to the sum of squares on the remaining two sides.
Mathematically, (perpendicular}^{2} + (base)^{2} = (hypotenuse)^{2} [i.e. p^{2} + b^{2} = h^{2}]
For example:
If perpendicular (p) = 4cm, base (b) = 3 cm, then hypotenuse (h) = ?
Solution:
Perpendicular (p) = 2 cm
Base (b) = 3 cm
Hypotenuse (h) = ?
By formulae
h^{2}= p^{2} + b^{2}
or, h^{2} =4^{2} + 3^{2}
or, h^{2} = 16 + 9
or, h^{2} = \(\sqrt{25}\)
\(\therefore\) h = 5 cm.
Note: h^{2}= p^{2} + b^{2} p^{2} = h^{2} - b^{2} , p = \(\sqrt{h^2 - b^2}\) b^{2}= h^{2}- p^{2}, b = \(\sqrt{h^2 - p^2}\) |
Converse of Pythagoras Theorem
As we know that in a right-angled triangle , h^{2}= p^{2} + b^{2} which is known as Pythagoras theorem .On the contrary, the converse of Pythagoras relation holds true i.e, if h^{2}= p^{2} - b^{2} holds true then the triangle is the right angled . If the relation is false , the triangle is not theright-angled triangle . For example:
Given that hypotenuse (h) = 4 cm, base (b) = 3 cm and perpendicular (p) = 2 cm. Whether it is true or false?
Solution:
Hypotenuse (h) = 4 cm
Base (b) = 3 cm
Perpendicular (p) = 2 cm
By formulae,
h^{2} = p^{2} + b^{2}
or, (4cm)^{2} = (2cm)^{2} + (3cm)^{2}
or, 16cm^{2} = 4cm^{2} + 9cm^{2}
or 16cm^{2} = 13cm^{2}, which is false.
Solution ;
Here, ΔABC is a right angled triangle right angled at A.
i.e. ∠BAC = 90 .
∠ACB = α ( reference angle )
∴ Side AB = perpendicular ( p )
Side BC = Hypotenuse ( h )
Side AC =base ( b )
Pythagoras Theorem is a relation developed on the basis of the elements of right angled triangle. It is established by a famous mathematician named Pythagoras. The elements of a right-angled triangle are perpendicular, base and hypotenuse.
Solution ;
Here, ΔQRS is a right angled triangle right angled at R.
i.e. ∠QRS = 90 .
∠QSR = α ( reference angle )
∴ Side QR = perpendicular ( p )
Side QS = Hypotenuse ( h )
Side SR =base ( b )
Again,
we know that,
h^{2} = p^{2} + b^{2 }
h^{2} = (3 cm)^{2 }+ (4 cm)^{2}
h^{2} = 9 cm + 12 cm
h^{2} = 25 cm
h^{2} = (5 cm)^{2}
h = 5
Solution
Here, in the right-angled Δ ABC = 90
So, h^{2} = P^{2} + b^{2}
(AB)^{2} = (AC)^{2} + (BC)^{2}
(AB)^{2} = (5cm)^{2} + (12 cm)^{2}
(AB)^{2} = 25 cm^{2} + 144 cm^{2}
^{ }(AB)^{2} = 169 cm^{2}
^{ }AB = 13 cm
Now, taking θ as the reference angle,
sinθ = \(\frac{p}{h}\)
= \(\frac{BC}{AB}\) = \(\frac{3}{4}\)
Again, takiing ∝ as the reference angle,
cos∝ = \(\frac{b}{h}\)
= \(\frac{BC}{AB}\) = \(\frac{3}{4}\)
Solution
Here, in the right-angled Δ ABC = 90
So, h^{2} = P^{2} + b^{2}
(AB)^{2} = (AC)^{2} + (BC)^{2}
(AB)^{2} = (25cm)^{2} + (12 cm)^{2}
(AB)^{2} = 25 cm^{2} + 144 cm^{2}
^{ }(AB)^{2} = 169 cm^{2}
^{ }(AB)^{2} = (13 cm)^{2}
^{ }AB = 13 cm.
Solution
Here, in the right-angled Δ ABC = 90
So, h^{2} = P^{2} + b^{2}
(AB)^{2} = (AC)^{2} + (BC)^{2}
(AB)^{2} = (25cm)^{2} + (12 cm)^{2}
(AB)^{2} = 25 cm^{2} + 144 cm^{2}
^{ }(AB)^{2} = 169 cm^{2}
^{ }(AB)^{2} = (13 cm)^{2}
^{ }AB = 13 cm.
Solution
We know ,
h^{2} = p^{2 } + b^{2}( Since ΔABC is a right angled triangle)
(AB)^{2} = (AB)^{2} + (BC)^{2}
(AB)^{2} = (2cm)^{2} + (2cm)^{2}
(AB)^{2} = 4cm^{2}
AB = 4 cm ans.
The side opposite to right angle is called _____ .
The side opposite to reference angle is known as _____ .
The side except hypotenuse and the perpendicular is called_____.
Hypotenuse is denoted by _____ .
Base is denoted by _____ .
Perpendicular is denoted by _____ .
A triangle whose all angles are less than 90 is called _____ .
A triangle having one angle 90 is called _____ .
A triangle whose one angle is 90 is known as a _____ .
A triangle with all 3 sides equal is known as an _____ .
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