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, 4.5, 3.2 a = □° (do not round until the final answer. then round to the nearest degree as needed.) b = □° (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.)

Explanation:

Step1: Define triangle side lengths

When circles touch externally, distance between centers = sum of radii. Let radii: $r_A=5.4$, $r_B=4.5$, $r_C=3.2$.
Sides:
$BC = r_B + r_C = 4.5 + 3.2 = 7.7$
$AC = r_A + r_C = 5.4 + 3.2 = 8.6$
$AB = r_A + r_B = 5.4 + 4.5 = 9.9$

Step2: Calculate angle A with Law of Cosines

Law of Cosines: $\cos A = \frac{AB^2 + AC^2 - BC^2}{2 \cdot AB \cdot AC}$
Substitute values:
$\cos A = \frac{9.9^2 + 8.6^2 - 7.7^2}{2 \cdot 9.9 \cdot 8.6} = \frac{98.01 + 73.96 - 59.29}{170.28} = \frac{112.68}{170.28} \approx 0.6617$
$A = \arccos(0.6617) \approx 48.5^\circ$

Step3: Calculate angle B with Law of Cosines

$\cos B = \frac{AB^2 + BC^2 - AC^2}{2 \cdot AB \cdot BC}$
Substitute values:
$\cos B = \frac{9.9^2 + 7.7^2 - 8.6^2}{2 \cdot 9.9 \cdot 7.7} = \frac{98.01 + 59.29 - 73.96}{152.46} = \frac{83.34}{152.46} \approx 0.5467$
$B = \arccos(0.5467) \approx 56.9^\circ$

Step4: Calculate angle C with angle sum

Triangle angle sum = $180^\circ$
$C = 180^\circ - A - B = 180 - 48.5 - 56.9 = 74.6^\circ$

Answer:

A = 49°
B = 57°
C = 75°

(Note: Rounded to nearest degree as required.)