Isosceles right triangle:-
Today in this article i will discuss on the topic isosceles right triangle,hope this will help a lot for school going children.To start my discuss we will first know,
**what is an isosceles triangle?
Answer:- A triangle whose any two sides are equal i.e; equal in measure is termed as isosceles triangle.
In the above picture, ABC is an isosceles Δ
whose AB=AC .now we will know,
***What is right triangle?
Answer:- A triangle whose any one angle measure 90° is known as right triangle.
finally we will learn,
**what is isosceles right triangle?
Answer:- A triangle, whose any two sides are equal and have an angle measure 90°,
is called isosceles right triangle.
In the above picture,
Δ PQR is an isosceles right triangle.
whose, base (QR) = perpendicular (PQ)
Illustration:-
A right triangle has three specified sides and they are hypotenuse,base and perpendicular with different measures and are denoted by h,b and p
but, in case of isosceles right triangle perpendicular and base are always equal.
I will discuss on area of isosceles right triangle.
We know,
Area of right triangle is = ½ .b.p
where, b = base of the right triangle
p = perpendicular of the right triangle.
But in case, of isosceles right triangle.
Area of isosceles right triangle is = ½ .b.p
= ½ b.b [∵In case of isosceles right triangle,perpendicular(p) =base(b)]
=½ b²
or,
= ½ .b.p
= ½ p.p
= ½ p² [∵In case of isosceles right triangle,perpendicular(p) =base(b)]
Some area problems of isosceles right triangle:-
Question(1):- The base of an isosceles right triangle is 4 cm, find its area.
Answer:-
∵ The base of isosceles right triangle(b)
= 4cm
∴ The area of the Δ =½ b²
= ½ .4²
= ½ .4.4
=½ .16
= 8cm²
Question(2):- The height or perpendicular of an isosceles right triangle is 2 m, find its area.
Answer:-
∵ The height of isosceles right triangle(p)
= 2 m
∴ The area of the Δ =½ p²
= ½.2²
= ½ .2.2
= 2 cm²
Question(3):- The area of an isosceles right triangle is 32 cm², find its all sides.
Answer:-
1st method,
The area of the isosceles right triangle
= 32 cm²
Let, the base of the triangle = b cm
∴ b = √2.area
= √2.32
=√64
= 8 cm
∴ base(b) = perpendicular(p) = 8 cm
[ In,case of isosceles right triangle both are always equal]
Let, hypotenuse of the Δ = h cm
Now applying Pythagorus law,we get
h²= b² +p²
= 8² + 8²
= 2.8²
∴ h = √2.8²
= 8√2 cm
2nd method:-
The area of the isosceles right triangle
= 32 cm²
Let, the base of the triangle = b cm
Now,
According to question.
½ b.p = 32
=>½ b.b = 32 [ ∵In,case of isosceles right triangle, b=p]
=> b² = 2.32
=>b² = 64
=> b =√64
=> b = 8
∴ base(b) = perpendicular(p) = 8 cm
[ In,case of isosceles right triangle both are always equal]
Let, hypotenuse of the Δ = h cm
Now applying Pythagorus law,we get
h²= b² +p²
= 8² + 8²
= 2.8²
∴ h = √2.8²
= 8√2 cm
So, three sides of the isosceles right triangle are = 8cm,8cm,and 8√2 cm
Now , I want to discuss a little more on
Perimeter of isosceles right triangle:-
** What do you mean by perimeter of isosceles right triangle?
Answer:-
The perimeter of isosceles right triangle
is sum of its all sides but in case of isosceles right triangle,perpendicular(p) =base(b)
So, perimeter of isosceles right triangle
will be = h + b+ p [ where h= hypotenuse
p = perpendicular
and b = base]
= h + b + b
= h + 2b
or,
= h + p +p
= h + 2p
Some perimeter problems of isosceles right triangle:-
Question(1):- The base of an isosceles right triangle is 3 m and its hypotenuse is 5 m, find its perimeter.
Answer:- ∵ The base of an isosceles right
triangle(b) = 3 m
and hypotenuse (h) = 5 m
∴ perimeter of the Δ = h + 2b
= (5 + 2×3)m
= ( 5 + 6) m
= 11 m
Question(2):- Hypotenuse of an isosceles right triangle is 10 cm and its perimeter is 20 cm , find other two sides of the triangle.
Answer:-
1st method,
∵ Hypotenuse of the isosceles right
triangle(h) = 10 cm
perimeter of the Δ = 20 cm
∴ base (b) =( perimeter - h)/2
= (20-10)/2
= 10/2
= 5 cm
and perpendicular of the Δ (p)=5 [∵ In case of isosceles right triangle b is always equal perpendicular]
∴ other two sides of the triangle are 5 cm and 5 cm.
2nd method,
∵ Hypotenuse of the isosceles right
triangle(h) = 10 cm
perimeter of the Δ = 20 cm
Let, the base of the Δ = b cm
now, according to question,
h+ 2b =20
=>10 + 2b = 20
=> 2b = 20 -10
=> b = 10/2
=>b = 5
[Note:- Students are advised 1st realise the above process how to solve isosceles right triangle's perimeter and area problems and then do themselves]
FREQUENTLY ASKED QUESTIONS:-
What is the area formula of a triangle?
Answer: The area formula of a triangle = ½ × b × h where , b = base of the triangle and h = heigh/altitude of the triangle.
How many triangles can be drawn using three collinear points ?
Answer:- Using three collinear points we cann't draw any triangle,i.e; zero triangle can be drawn as the points lie in the same line so there is no altitude,so there will be no triangle formed.
How many obtuse angle can have in a triangle?
Answer :- Only one obtuse angle.
What is measure of all angles of an acute angled triangle
Answer :- All angles measures less than 90° i.e; all angles are acute angles.
What is another name of an equilateral triangle?
Answer:- Equiangular triangle.
What is the name of the triangle whose two sides are equal?
Answer:- Isosceles triangle.
What is the area formula of a triangle?
Answer: The area formula of a triangle = ½ × b × h where , b = base of the triangle and h = heigh/altitude of the triangle.
How many triangles can be drawn using three collinear points ?
Answer:- Using three collinear points we cann't draw any triangle,i.e; zero triangle can be drawn as the points lie in the same line so there is no altitude,so there will be no triangle formed.
How many obtuse angle can have in a triangle?
Answer :- Only one obtuse angle.
What is measure of all angles of an acute angled triangle
Answer :- All angles measures less than 90° i.e; all angles are acute angles.
What is another name of an equilateral triangle?
Answer:- Equiangular triangle.
What is the name of the triangle whose two sides are equal?
Answer:- Isosceles triangle.
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