Question

the sides of a right angled triangle are consecutive positive integers find the area of the triangle​

Answer 1

Area of the triangle = 6 square units

Step-by-step explanation:

Given Data

The sides of a right angled triangle are consecutive integers.

To find- the area of the triangle

The Formula to find the area of a triangle is

$Area of a triangle = \frac{1}{2} \times base \times height$    (square units)

Let us consider following the positive integers x, (x + 1) and (x + 2) be the sides of right angled triangle

And consider the hypotenuse of triangle as (x + 2)

Pythagoras theorem  A² + B² = C²

Substitute the respective values in above formula

x² + (x + 1)² = (x + 2)²

x² = (x + 2)² – (x + 1)²

Let us take  (x + 2)² = a  and  (x + 1)² = b

Then use the  identity a² - b² =  (a + b)(a – b)

Substitute the values of 'a' and 'b'

x² = (x + 2 + x + 1)(x + 2 – x – 1)

x² = (2x + 3)(1)

x² = 2x + 3

x² – 2x – 3 = 0  

x² – 3x + x – 3 = 0

The above expression can also be written as  

x(x – 3) + 1(x – 3) = 0  

(x + 1)(x – 3) = 0

(x + 1) = 0 or (x – 3) = 0

x = -1 or x = 3

Take the positive value, x = 3

If x= 3, then

x + 1 = (3 + 1)  = 4

And   x + 2 = (3 + 2) = 5

Then the three sides of triangle are 3, 4 and 5

For a right angled triangle one side would be the base and the other as height

We took (x+5) as hypotenuse which is 5

Then Base = 3 and height = 4  

$Area of a triangle = \frac{1}{2} \times base \times height$

$Area of triangle = \frac{1}{2} \times 3 \times 4 = \frac{1}{2} \times 12$

Area of triangle = 6 Square units

Therefore the area of right angled triangle whose sides are consecutive integers(3, 4 and 5) is 6 square units  

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