Given BC = 53 cm, BD = 62 cm, CD = 80 cm, ABC = 53°, and ACB = 66°, find the following. (Round your answers to the nearest whole number.)
(a) the length of the chainstay, AC
AC = cm
(b) BCD
BCD = °
where's D? is ABC a triangle?
To find the length of the chainstay, AC, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2abcos(C)
In this case, we have BC = 53 cm, BD = 62 cm, and ABC = 53°. Let's substitute these values into the equation:
AC^2 = BC^2 + BD^2 - 2BC * BD * cos(ABC)
AC^2 = 53^2 + 62^2 - 2 * 53 * 62 * cos(53)
Now we can calculate AC by taking the square root of both sides:
AC = √(53^2 + 62^2 - 2 * 53 * 62 * cos(53))
Use a calculator to compute the square root and evaluate the expression. Round the answer to the nearest whole number to find the length of the chainstay, AC.
For part (b), to find the angle BCD, we can use the Law of Sines. The Law of Sines states that in a triangle with sides a, b, and c, and angles A, B, and C, the following equation holds:
a/sin(A) = b/sin(B) = c/sin(C)
In this case, we want to find angle BCD. We are given BC = 53 cm, CD = 80 cm, and AC = length of the chainstay found in part (a). Let's use these values to find angle BCD:
BC/sin(BCD) = CD/sin(ABC)
Substitute the known values:
53/sin(BCD) = 80/sin(53)
Now, solve for sin(BCD):
sin(BCD) = (53 * sin(53)) / 80
To find angle BCD, take the inverse sine of this value:
BCD = sin^(-1)((53 * sin(53)) / 80)
Evaluate the expression using a calculator and round the answer to the nearest whole number to find angle BCD.