Lesson Three circles touching externally
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<H2>Three circles touching externally</H2> <H3>Problem 1</H3>Two circles with centers A and B are touching externally. Third circle with center C touches both the circles A and B externally. Suppose AB = 3cm, BC = 3cm and AC = 4cm. Find the radii of the circles. <B>Solution</B> <pre> When two circles touch externally, the distance between their center is the sum of their radii. It gives you the system of three equations a + b = 3, (1) a + c = 4, (2) b + c = 3. (3) where a, b and c are the radii of the three circles with the centers A, B and C respectively. To solve the system, first add all the three equations. You will get 2a + 2b + 2c = 3 + 4 + 3 = 10, which implies a + b + c = 5. (4) Now subtract equation (3) from equation (4) (both sides). You will get a = 5-3 = 2. Subtract equation (2) from equation (4) (both sides). You will get b = 5-4 = 1. Finally, subtract equation (1) from equation (4) (both sides). You will get a = 5-3 = 2. <U>Answer</U>. The radii are: 2 cm for the circle with the center at A; 1 cm for the circle with the center at B; 2 cm for the circle with the center at C. </pre> <H3>Problem 2</H3>The centers of three circles that are tangent to each other are connected to form a triangle that has side measurements of 4cm, 6cm, and 8cm. Find the radius of the biggest circle. <B>Solution</B> <pre> Let the radii of these circles be "a", "b" and "c", in increasing order. Then you have these equations a + b = 4, (1) a + c = 6, (2) b + c = 8. (3) Add the three equations, (1),(2) and (3), You will get 2a + 2b + 2c = 4 + 6 + 8 = 18, which implies a + b + c = 9. (4) Now subtract eq(1) from eq(4). You will get c = 9-4 = 5. Next, subtract eq(2) from eq(4). You will get b = 9-6 = 3. Finally, subtract eq(3) from eq(4). You will get a = 9-8 = 1. Thus you found the radii. They are a= 1, b= 3 and c= 5. 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