Answer:
Explanation:
To find the length of segment CD, we can use the law of cosines. The law of cosines states that for a triangle with sides a, b, and c, and an angle opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, we can label the sides of triangle CAD as follows:
a = AC (radius of circle A) = 6in
b = CD (length we want to find)
c = AD (unknown)
We also know that angle CAD = 72°. Plugging in these values into the law of cosines, we get:
AD^2 = 6^2 + CD^2 - 2*6*CD*cos(72°)
Now, let's move on to circle B and triangle CBD. Again, we can label the sides:
a = BC (radius of circle B) = 8in
b = CD (length we want to find)
c = BD (unknown)
We also know that angle CBD = 54°. Using the law of cosines, we get:
BD^2 = 8^2 + CD^2 - 2*8*CD*cos(54°)
Now, we have two equations with two unknowns (AD and BD). However, we are only interested in the length of CD, which is common to both equations.
To solve for CD, we can set the two equations equal to each other:
6^2 + CD^2 - 2*6*CD*cos(72°) = 8^2 + CD^2 - 2*8*CD*cos(54°)
Simplifying and rearranging the equation, we get:
36 - 96*cos(72°) = 64 - 128*cos(54°)
Now, we can solve for CD:
CD = (36 - 96*cos(72°) - 64 + 128*cos(54°)) / (2*cos(72°) - 2*cos(54°))
Calculating the values on the right side of the equation, we find:
CD ≈ 1.87 inches
Therefore, the length of segment CD is approximately 1.87 inches.