Question

The three circles are arranged so that they touch each other, as shown in the

figure. Use the given radii for the circles with centers A, B, and C, respectively,
to solve triangle.
5.4, 4.4, 3.4
***
A=
□°
(Do not round until the final answer. Then round to the nearest degree as needed.).
B = 0°
(Do not round until the final answer. Then round to the nearest degree as needed.)
c=
(Do not round until the final answer. Then round to the nearest degree as needed.)
B

Answer 1

Explanation:

In the given figure, we have three circles arranged such that they touch each other. Let the centers of these circles be A, B, and C, with radii 5.4, 4.4, and 3.4, respectively.

We can see that triangle ABC is an equilateral triangle, since all sides are of equal length (the radii of the circles).

To find the angle A, we can use the law of cosines, which states that:

c^2 = a^2 + b^2 - 2ab cos(C)

where a, b, and c are the lengths of the sides of a triangle, and C is the angle opposite the side of length c.

Since triangle ABC is equilateral, we have a = b = c, and C = 60°. Therefore, we can rewrite the above equation as:

c^2 = 2a^2 - 2a^2 cos(60°)

Simplifying and solving for a, we get:

a = c / sqrt(3)

Substituting the given values, we have:

a = 4.4 / sqrt(3) ≈ 2.54

Therefore, angle A is:

A = 180° - 60° - 60° = 60°

And angle B is:

B = 180° - A = 120°

Finally, we can use the law of sines to find the length of side c:

sin(A) / a = sin(B) / b = sin(C) / c

Substituting the values we have found, we get:

sin(60°) / 2.54 = sin(120°) / c

Simplifying and solving for c, we get:

c = 2.54 / sqrt(3) / sin(120°) ≈ 3.71

Therefore, the length of side c is approximately 3.71, and angle B is 120°.