the radius of the incircle of a triangle is 2cm. if the area of the triangle is 6 sq.cm, then its perimeter is.
area = inradius * perimeter
6 = 2p
p=3
The formula comes from the fact that the incenter is the intersection of the angle bisectors. Draw altitudes from the incenter, and each of the 6 interior triangles has base 1/2 of a side, and altitude = inradius.
To find the perimeter of the triangle, we need to first determine the lengths of its sides.
We know that the radius of the incircle is 2 cm. The radius of the incircle is also the inradius of the triangle.
The formula to find the inradius given the area and the semiperimeter (s) of the triangle is:
inradius = Area / s
Let's denote the sides of the triangle as a, b, and c. Then the semiperimeter (s) is:
s = (a + b + c) / 2
Since the inradius is given as 2 cm and the area is given as 6 sq.cm, we can rewrite the formula as:
2 = 6 / [(a + b + c) / 2]
Simplifying the equation, we obtain:
4 = 6 / (a + b + c)
(a + b + c) = 6 / 4
(a + b + c) = 3/2
Now, we can use this information to find the lengths of the sides of the triangle.
We also know that the area of the triangle is given by the formula:
Area = (s*(s-a)*(s-b)*(s-c))^(1/2)
Substituting the value of s from the previous equation into the area formula, we have:
6 = (√[(3/2)*((3/2)-a)*((3/2)-b)*((3/2)-c)])
Simplifying further, we get:
6 = (√[(9/4)-[(3/2)*(a+b+c)]/2+(1/4)*(ab+bc+ac)])
Rearranging terms, we obtain:
36 = (9-(3/2)(a+b+c)+(1/2)(ab+bc+ac))
Multiplying through by 4, we have:
144 = 36-6(a+b+c)+(ab+bc+ac)
Rearranging again, we get:
6(a+b+c)-(ab+bc+ac) = 108
Now, we have two equations:
(a + b + c) = 3/2
6(a+b+c)-(ab+bc+ac) = 108
Using these equations, we can solve for the side lengths of the triangle.
Unfortunately, the equations are not sufficient to find the specific side lengths without more information. Therefore, we cannot determine the perimeter of the triangle at this time.
To find the perimeter of the triangle, we need to know the lengths of the sides. However, we're given the radius of the incircle and the area of the triangle. We can use these pieces of information to find the lengths of the sides of the triangle.
First, let's understand the relationship between the radius of the incircle, the semi-perimeter of the triangle, and the area of the triangle.
The radius of the incircle is given by the formula:
radius = area / semi-perimeter
We are given that the radius is 2 cm and the area is 6 sq. cm. Substituting these values into the formula, we can solve for the semi-perimeter.
2 cm = 6 sq. cm / semi-perimeter
To find the semi-perimeter, we can multiply both sides of the equation by the semi-perimeter:
2 cm * semi-perimeter = 6 sq. cm
Dividing both sides by 2 cm, we get:
semi-perimeter = 6 sq. cm / 2 cm
semi-perimeter = 3 cm
Now, we know that the semi-perimeter is 3 cm. The perimeter of a triangle is equal to twice the semi-perimeter:
perimeter = 2 * semi-perimeter
Substituting the value of the semi-perimeter, we get:
perimeter = 2 * 3 cm
perimeter = 6 cm
Therefore, the perimeter of the triangle is 6 cm.