## 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

## 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.