three circles of radie 3,4, and 5 units respectively are tangent to each other eternaly. find the angle of the triangle formed by joining centers?
help plz
so the triangle has sides, 7, 8 and 9
find the angle opposite the 9 side first, calling it Ø
and using the cosine law.
9^2 = 7^2 + 8^2 - 2(7)(8)cosØ
81 = 49+64 -112cosØ
cosØ = 32/112
Ø = 73.398..° (I stored that in my calculator memory for future accuracy)
let x be the angle opposite the side 7
sinx/7 = sin73.398/9
sinx = .745356
x = 48.19°
so the third angle is 180-48.19-73.398 = 58.41°
thanks sir
you are welcome
little trick...
when finding the angles in a triangle given the 3 sides, you will of course have to use the cosine law to find one angle, then you can use the sine law to find a second angle.
Always find the largest angle first using the cosine law.
That way if the angle is obtuse, that problem is out of the way.
Using the sine law, gives you two possible answers, one acute and one obtuse.
Since a triangle can have at most only one obtuse angle, if you found it using the cosine law, the ambiguous case of the two answers is eliminated.
To find the angle of the triangle formed by joining the centers of the three circles, we can use trigonometry and the properties of tangents.
Let's label the centers of the circles as A, B, and C. Since the circles are tangent externally, the radii drawn to the points of tangency will be perpendicular to the tangent lines. This forms a right-angled triangle within each circle.
Let's consider the triangle formed by joining the centers A, B, and C. To find the angle at A, we can use the concept of the inverse tangent (arctan), along with the known lengths of the radii.
First, let's find the side lengths of the triangle:
- The distance between A and B is 3+4 = 7 units.
- The distance between A and C is 3+5 = 8 units.
- The distance between B and C is 4+5 = 9 units.
Next, we can use the cosine rule, which relates the side lengths of a triangle to its angles, to find the angle at A:
cos(A) = (b^2 + c^2 - a^2) / (2bc)
In this case:
- a = 8 units (distance between A and C)
- b = 7 units (distance between A and B)
- c = 9 units (distance between B and C)
Plugging these values into the formula, we have:
cos(A) = (7^2 + 9^2 - 8^2) / (2 * 7 * 9)
Calculating the value on the right-hand side of the equation:
cos(A) = (49 + 81 - 64) / 126
cos(A) = 66 / 126
cos(A) = 11 / 21
Now, we can find the angle at A by taking the inverse cosine (arccos) of the value obtained:
A = arccos(11 / 21)
Using a calculator, we find:
A ≈ 58.61°
Therefore, the angle of the triangle formed by joining the centers of the three circles is approximately 58.61 degrees.