Please solve The radius of the circle inscribed in the triangle formed by lines X=0 , y =0 , 4x 3y-24=0 =?

Question

Answer

Problem:

The radius of the circle inscribed in the triangle formed by lines X=0 , y =0 , 4x 3y-24=0 =?

Solution:

Introduction:

To find the radius of a circle inscribed in a triangle, we need to use the formula:

r = A / s

where r is the radius of the inscribed circle, A is the area of the triangle, and s is the semiperimeter of the triangle.

Steps:

Step 1:

Identify the vertices of the triangle formed by the given equations.

From the given equations, we can find the three vertices of the triangle:

Vertex 1: (0,0)

Vertex 2: (8,0) (Solving 4x+3y=24 for x when y=0)

Vertex 3: (0,8) (Solving 4x+3y=24 for y when x=0)

Step 2:

Calculate the lengths of the sides of the triangle.

We can use the distance formula to find the lengths of the sides:

Side 1: AB = 8

Side 2: AC = 8

Side 3: BC = 8√2

Step 3:

Calculate the area of the triangle.

We can use Heron's formula to find the area of the triangle:

s = (AB + AC + BC)/2 = 12 + 4√2

A = √(s(s-AB)(s-AC)(s-BC)) = 32

Step 4:

Calculate the semiperimeter of the triangle.

The semiperimeter of the triangle is half the perimeter:

s = (AB + AC + BC)/2 = 12 + 4√2

Step 5:

Calculate the radius of the inscribed circle.

Using the formula r = A / s, we get:

r = 32 / (12 + 4√2) = 4(2 - √2)

Answer:

The radius of the circle inscribed in the triangle formed by lines X=0 , y =0 , 4x 3y-24=0 is 4(2 - √2).

Answer

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