three circles of radii 27,38, and 42 centimeter respectively are tangent to each other. find the median to the longest side of the triangle formed by joining the centers
The sides of the triangle are 65,69,80 (sums of radii).
So, you can probably find the medians given the sides, yes?
To find the median to the longest side of the triangle formed by joining the centers of the three tangent circles, let's start by drawing a sketch to visualize the problem.
We have three circles with radii 27, 38, and 42 centimeters, respectively. Let's label the centers of these circles as A, B, and C, respectively.
Since the circles are tangent to each other, we can draw lines from each center to the points of tangency, forming an equilateral triangle ABC.
Now, let's label the points of tangency on the circles. The points where the circle with radius 27 touches the other two circles will be D and E, respectively. The points where the circle with radius 38 touches the other two circles will be F and G, respectively. The points where the circle with radius 42 touches the other two circles will be H and I, respectively.
Connecting points D, F, and H will form the longest side of the equilateral triangle ABC.
To find the median to the longest side, we need to know the length of the longest side.
Using the property of equilateral triangles, we know that the side length of the triangle formed by joining the centers of the tangent circles is equal to the sum of the radii of the three circles.
Side length of equilateral triangle ABC = 27 + 38 + 42 = 107 centimeters
Now, let's find the length of the longest side DH using Pythagoras' theorem.
DH is the hypotenuse of right-angled triangle ADH.
From the right-angled triangle ADH, we can see that the height (HD) will be 27 + 38 = 65 centimeters (the sum of radii of circles with radii 27 and 38).
Using Pythagoras' theorem, we can find the length of DH:
DH^2 = (AD^2 + HD^2)
The length of AD is the side length of the equilateral triangle ABC divided by 2, which is 107/2 = 53.5 centimeters.
Plugging in the values:
DH^2 = (53.5^2 + 65^2)
DH^2 = (2862.25 + 4225)
DH^2 = 7087.25
Taking the square root of both sides:
DH ≈ 84.19 centimeters
Therefore, the median to the longest side DH in the triangle formed by joining the centers of the tangent circles is approximately 84.19 centimeters.