A circle of radius 2 is externally tangent to a circle of radius 8. Compute the length of their common tangent.
To find the length of the common tangent, we first need to understand the geometry of the circles.
In this case, we have two circles, one with a radius of 2 and another with a radius of 8. Let's call the center of the smaller circle A and the center of the larger circle B.
From the given information, we know that the two circles are externally tangent, which means they touch each other at one point. Let's call this point C.
To find the length of the common tangent, we can draw a line segment from point C to the center of the larger circle, point B.
Now, we have a right triangle ABC, where AB is the radius of the larger circle (8), BC is the radius of the smaller circle (2), and AC is the length of the common tangent that we want to find.
Using the Pythagorean theorem, we can find the length of AC:
AC^2 = AB^2 - BC^2
AC^2 = 8^2 - 2^2
AC^2 = 64 - 4
AC^2 = 60
Taking the square root of both sides, we find:
AC = sqrt(60)
Simplifying, we get:
AC = 2 * sqrt(15)
Therefore, the length of the common tangent is 2 * sqrt(15).
To compute the length of the common tangent between two circles, we can use a property of tangents. A tangent is perpendicular to the radius of a circle at the point of tangency.
In this case, the smaller circle with radius 2 is externally tangent to the larger circle with radius 8. The centers of both circles are along the line connecting the centers of the circles.
To find the length of the common tangent, we first need to find the distance between the centers of the circles. Since the smaller circle is externally tangent to the larger circle, the distance between their centers is equal to the sum of their radii.
The radius of the smaller circle is 2, and the radius of the larger circle is 8. Therefore, the distance between their centers is 2 + 8 = 10.
Next, we form a right triangle between the center of the larger circle, the center of the smaller circle, and the point of tangency. The distance between the centers (10) forms the hypotenuse of the right triangle.
We can use the Pythagorean theorem to find the length of the segment connecting the center of the larger circle and the point of tangency, which is a radius of the larger circle. Let's call this length 'x'.
Using the Pythagorean theorem, we have:
x^2 + 2^2 = 10^2
x^2 + 4 = 100
x^2 = 96
x = sqrt(96)
x ≈ 9.8
Therefore, the length of the common tangent between the circles is approximately 9.8 units.
Draw radii to the points of tangency.
Draw a line connecting the centers.
Draw a parallel line, 2 units closer to the tangent points.
Now you have a right triangle with legs 10 and 6, with the tangent line forming the hypotenuse.