Three circles touch one another externally.Their centres form A triangle with sides 10cm,9cm and 7cm.Find the radii of the circles
this is the other problem backwards. You need to find three numbers which add up in pairs to the given sides.
To find the radii of the circles, we need to use a property of circles that states that the tangent to a circle is perpendicular to the radius at the point of contact.
Let's denote the radii of the circles as r1, r2, and r3.
Now, let's consider the triangle formed by the centers of the circles. According to the triangle inequality theorem, the sum of any two sides of a triangle must be greater than the third side.
Using this property, we can write the following inequalities:
r1 + r2 > 10 (1)
r2 + r3 > 9 (2)
r1 + r3 > 7 (3)
Now, since the circles touch each other externally, the point of contact between any two circles lies on the line connecting the centers of the circles. This means that the line connecting the centers of the circles is the perpendicular bisector of the common tangent between the circles.
From this information, we can form the following system of equations:
2r1 = 10 (4) (Since the side opposite to the longest side in a triangle is twice the radius of the circumcircle)
2r2 = 9 (5)
2r3 = 7 (6)
From equations (4), (5), and (6), we can find the values of r1, r2, and r3:
r1 = 10/2 = 5 cm
r2 = 9/2 = 4.5 cm
r3 = 7/2 = 3.5 cm
Therefore, the radii of the circles are 5 cm, 4.5 cm, and 3.5 cm.