Two circles with radii 3cm and r cm touch each other externally. If their common tangent is 8cm long,what is the value of r?
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To find the value of r, we will use the concept of the Pythagorean Theorem and the properties of tangents.
Let's assume that the center of the first circle is A and the center of the second circle is B. The radii of the first and second circles are 3 cm and r cm, respectively. Also, let's label the point of contact of the common tangent on the first circle as C and the point of contact on the second circle as D.
We know that the line connecting the centers of two circles is perpendicular to the common tangent at the point of contact.
Now, let's consider the right-angled triangle ABC. The sides of this triangle are the radius of the first circle (3 cm), the radius of the second circle (r cm), and the distance between the centers of the circles (which is the sum of the radii, 3 cm + r cm = 3 + r cm).
According to the Pythagorean Theorem, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is the common tangent (8 cm), and the other two sides are the radius of the first circle (3 cm) and the distance between the centers (3 + r cm).
Applying the Pythagorean Theorem, we can write the equation as:
8^2 = 3^2 + (3 + r)^2
Simplifying this equation:
64 = 9 + 9 + 6r + r^2
Combining like terms:
0 = r^2 + 6r - 46
To find the value of r, let's solve this quadratic equation by factoring, completing the square, or using the quadratic formula.
Factoring the quadratic equation or completing the square method is not easily apparent in this case, so let's use the quadratic formula:
r = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = 6, and c = -46. Substituting these values into the quadratic formula, we get:
r = (-6 ± √(6^2 - 4*1*(-46))) / (2*1)
Simplifying further:
r = (-6 ± √(36 + 184)) / 2
r = (-6 ± √220) / 2
Now, calculate the value of r by taking the positive and negative square root of the discriminant and simplifying:
r = (-6 + √220) / 2 ≈ 4.29 cm (approximately)
r = (-6 - √220) / 2 ≈ -10.29 cm (approximately)
Since we are dealing with measurements of length, the negative value of r is not meaningful in this context. Therefore, the value of r is approximately 4.29 cm.