Two circles have a radii of 15 and 95. If the two external tangents to the circles intersect at 60 degrees, how far apart are the centers of the circles?
Can someone please explain this to me, show the work and give me the answer. Thanks!
As usual, draw a diagram. If the smaller circle has center O and the larger circle has center P, and the tangent on one side touches O at A and P at B, then draw AC parallel to OP, intersecting PB at Q.
Note that AQ = OP and you have a right triangle AQB where
angle QAB = 30°
QB = 80
So, the distance between the centers, OP = AQ = 80√3
To find the distance between the centers of the circles, we can form a right triangle with the radii of the circles as the legs and the distance between the centers as the hypotenuse.
Let's call the distance between the centers of the circles as "d."
Since the two external tangents to the circles intersect at 60 degrees, we can form a triangle with the radii and a side connecting the centers, with a 60-degree angle.
Using trigonometric ratios, we can find the value of "d."
In a right triangle with a 60-degree angle, the side opposite the angle is equal to half the hypotenuse. Therefore, the distance between the centers, "d," can be expressed as:
d = 2 * (radius1 * sin(60 degrees))
Given that the radius of the first circle is 15, we can substitute this value into the equation:
d = 2 * (15 * sin(60 degrees))
To find the value of sin(60 degrees), we can use the sine function:
sin(60 degrees) = 0.866
Now we can substitute this value into the equation:
d = 2 * (15 * 0.866)
d = 2 * 25.98
d = 51.96
Therefore, the distance between the centers of the circles is approximately 51.96 units.
To solve this problem, we can use the concept of right triangles formed by the centers of the circles, the distance between their centers, and the line connecting the centers.
Step 1: Draw a diagram of the problem to better visualize the given information. Label the centers of the circles as point A and point B. Draw the two external tangents of the circles, and mark the point of intersection as point C.
Step 2: Note that the radii of the circles are given as 15 and 95. The line connecting the centers (AB) forms a right triangle with the radii of the circles as the legs. Let AC represent one radius (15 units), and BC represent the other radius (95 units).
Step 3: From the given information, we know that the external tangents intersect at an angle of 60 degrees at point C. This means that angle ACB is a right angle (90 degrees - 60 degrees).
Step 4: Determine the length of the side AB, which represents the distance between the centers of the circles. To find AB, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (AB) is equal to the sum of the squares of the other two sides (AC and BC).
Applying the Pythagorean theorem, we have:
AB^2 = AC^2 + BC^2
AB^2 = 15^2 + 95^2
AB^2 = 225 + 9025
AB^2 = 9250
Step 5: Take the square root of both sides of the equation to find the length of AB:
AB = sqrt(9250)
AB ≈ 96.13
Therefore, the distance between the centers of the circles is approximately 96.13 units.