Three circles which touch each other externally with the center of the circles at A, B, C form a triangle with size of 9cm, 7cm, 6cm.calculate the radii of the circle
each side of the triangle is the sum of two radii.
To find the radii of the circles, we can use a property of circles that states that the radius drawn to a point of tangency is perpendicular to the tangent.
Let's denote the radii of the circles as r₁, r₂, and r₃, respectively. We will consider the right triangle formed by the centers of the circles (A, B, C) and the points where the circles touch.
Step 1: Draw radii from the center of each circle to the points of tangency.
Step 2: Notice that the radii (r₁, r₂, and r₃) are perpendicular to the sides of the triangle formed by the centers of the circles (A, B, C).
Step 3: Use the Pythagorean theorem to find the lengths of the sides of the triangle:
Side AB: r₁ + r₂ = 9 cm (1)
Side AC: r₁ + r₃ = 7 cm (2)
Side BC: r₂ + r₃ = 6 cm (3)
Step 4: We now have a system of three equations (1, 2, 3) with three unknowns (r₁, r₂, r₃).
Solving the system of equations will give us the values of the radii.
By solving equations (1) and (3), we get:
r₁ + r₂ = 9 cm
r₂ + r₃ = 6 cm
Subtract equation (3) from equation (1):
r₁ - r₃ = 3 cm (4)
Adding equation (4) to equation (2):
2r₁ = 10 cm
r₁ = 5 cm
Substituting r₁ = 5 cm in equation (1):
5 cm + r₂ = 9 cm
r₂ = 4 cm
Substituting r₁ = 5 cm in equation (2):
5 cm + r₃ = 7 cm
r₃ = 2 cm
Therefore, the radii of the three circles are:
r₁ = 5 cm
r₂ = 4 cm
r₃ = 2 cm