In a right triangle with sides of lengths a, b and c (where c is the length of the hypotenuse), show that the length of the radius of the inscribed circle is
The length of radius of circle inscribed in a right angled triangle is .
Area of a triangle with an inscribed circle:
If P is the perimeter of the triangle and r is the length of radius of its inscribed circle, then the area A of the triangle is given by
Area of triangle:
If a, b and c are lengths of sides of triangle, then area of triangle is given by the formula:
Where s is the semi perimeter which is given by .
In a right angled triangle, if the length of hypotenuse is c and the length of remaining two sides of triangle is a and b, then by Pythagoras theorem, .
Expansion of :
After expansion, the above expression can be written as .
Expansion of :
Expansion of :
To find the radius of inscribed circle, we will use the following formula:, where A is the area of triangle, P is the perimeter of triangle and r is the radius of inscribed circle.
So, let’s find the area and perimeter of right angled triangle.
The lengths of sides of triangle is given as , and
Perimeter of triangle is sum of lengths of all sides of triangle.
Thus, perimeter .
Let A be the area of triangle.
The semi perimeter is half of the perimeter of triangle. Thus, .
Let’s substitute these values in the formula to find area of triangle.
Rearrange the products to simplify:
Using the expansion formula, can be written as:
can be further simplified as
It is given that c is the hypotenuse of the right angled triangle
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