Question

determine the remaining sides and angles of the triangle abc.
a = 200 m, a = 36° 48, c = 29° 32
what is the measure of angle b?
b = □°□
what is the length of side b?
b = □ m
(do not round until the final answer. then round to the nearest meter as needed.)
what is the length of side c?
c = □ m
(do not round until the final answer. then round to the nearest meter as needed.)

Explanation:

Step1: Convert angles to decimal degrees

First, convert \( A = 36^\circ 48' \) and \( C = 29^\circ 32' \) to decimal degrees.
For \( A \): \( 48'=\frac{48}{60}=0.8^\circ \), so \( A = 36.8^\circ \).
For \( C \): \( 32'=\frac{32}{60}\approx0.5333^\circ \), so \( C \approx 29.5333^\circ \).

Step2: Find angle B

In a triangle, \( A + B + C = 180^\circ \), so \( B = 180^\circ - A - C \).
Substitute the values: \( B = 180 - 36.8 - 29.5333 = 113.6667^\circ \).
Convert \( 0.6667^\circ \) to minutes: \( 0.6667\times60\approx40' \), so \( B = 113^\circ 40' \).

Step3: Use the Law of Sines to find side b

The Law of Sines is \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \).
We know \( a = 200 \), \( A = 36.8^\circ \), \( B = 113.6667^\circ \).
So \( b=\frac{a\sin B}{\sin A} \).
Calculate \( \sin A=\sin(36.8^\circ)\approx0.6 \), \( \sin B=\sin(113.6667^\circ)=\sin(180 - 66.3333^\circ)=\sin(66.3333^\circ)\approx0.9107 \).
Then \( b=\frac{200\times0.9107}{0.6}\approx\frac{182.14}{0.6}\approx303.57 \approx 304 \) (rounded to nearest meter).

Step4: Use the Law of Sines to find side c

Using \( \frac{a}{\sin A}=\frac{c}{\sin C} \), so \( c=\frac{a\sin C}{\sin A} \).
\( \sin C=\sin(29.5333^\circ)\approx0.493 \).
Then \( c=\frac{200\times0.493}{0.6}\approx\frac{98.6}{0.6}\approx164.33 \approx 164 \) (rounded to nearest meter).

Answer:

• \( B = \boldsymbol{113^\circ 40'} \)
• \( b = \boldsymbol{304} \) m
• \( c = \boldsymbol{164} \) m