Question

use the information to answer the questions.
in triangle abc, a = 11 in., m∠b = 70°, and c = 9 in.
which information about the triangle is given?
sas
complete
which formula can you use to find b?
\\( \frac { a } { \sin ( a ) } = \frac { b } { \sin ( b ) } \\)
\\( c \square \sin ( b ) = b \square \sin ( a ) \\)
\\( b ^ { 2 } = a ^ { 2 } + c ^ { 2 } - 2 a c \square \cos ( b ) \\)
\\( b ^ { 2 } = a ^ { 2 } + c ^ { 2 } - 2 a c \square \cos ( b ) \\)
complete
what is b, rounded to the nearest tenth?
3.8 in
10.4 in
11.6 in
16.4 in

Explanation:

Step1: Substitute the values into the formula

We know the formula \(b^{2}=a^{2}+c^{2}-2ac\cos(B)\). Given \(a = 11\), \(c = 9\), and \(B=70^{\circ}\).
First, calculate \(a^{2}=11^{2}=121\), \(c^{2}=9^{2} = 81\), and \(2ac\cos(B)=2\times11\times9\times\cos(70^{\circ})\).
Since \(\cos(70^{\circ})\approx0.3420\), then \(2\times11\times9\times0.3420=2\times99\times0.3420 = 198\times0.3420=67.716\).

Step2: Calculate \(b^{2}\)

\(b^{2}=a^{2}+c^{2}-2ac\cos(B)=121 + 81-67.716\).
\(b^{2}=121+81 - 67.716=134.284\).

Step3: Find \(b\)

Take the square root of \(b^{2}\), \(b=\sqrt{134.284}\approx11.6\) (rounded to the nearest tenth).

Answer:

\(11.6\) in