Answer:
b = 6.55
m∠A = 35°
m∠C = 35°
Step-by-step explanation:
In geometry, it is convention to name the sides of a triangle using lowercase letters corresponding to the angles they are opposite. For example, the side opposite angle A is typically referred to as side 'a'.
To find the length of side b of triangle ABC, we can use the Law of Cosines:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Law of Cosines}}\\\\b^2=a^2+c^2-2ac \cos B\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a, b$ and $c$ are the sides.}\\\phantom{ww}\bullet\;\textsf{$B$ is the angle opposite side $b$.}\end{array}}[/tex]
In this case:
Substitute the values into the equation and solve for b:
[tex]b^2=4^2+4^2-2(4)(4) \cos 110^{\circ}\\\\b^2=16+16-32 \cos 110^{\circ}\\\\b^2=32-32 \cos 110^{\circ}\\\\b=\sqrt{32-32 \cos 110^{\circ}}\\\\b=6.55321635431...\\\\b=6.55[/tex]
Therefore, the length of side b rounded to the nearest hundredth is b = 6.55.
[tex]\dotfill[/tex]
To find the measure of angle A, we can use the Law of Sines:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Law of Sines}} \\\\\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}\\\\\textsf{where:}\\\phantom{ww}\bullet \;\textsf{$A, B$ and $C$ are the angles.}\\\phantom{ww}\bullet\;\textsf{$a, b$ and $c$ are the sides opposite the angles.}\end{array}}[/tex]
In this case:
- B = 110°
- a = 4
- [tex]b=\sqrt{32-32 \cos 110^{\circ}}[/tex]
Substitute the values into the equation:
[tex]\dfrac{\sin A}{4}=\dfrac{\sin 110^{\circ}}{\sqrt{32-32 \cos 110^{\circ}}}[/tex]
Solve for A:
[tex]\sin A=\dfrac{4\sin 110^{\circ}}{\sqrt{32-32 \cos 110^{\circ}}}\\\\\\\\A=\sin^{-1}\left(\dfrac{4\sin 110^{\circ}}{\sqrt{32-32 \cos 110^{\circ}}}\right)\\\\\\\\A=35^{\circ}[/tex]
Note that by using the exact length of side b, angle A measures precisely 35°, eliminating the need for rounding.
As the interior angles of a triangle sum to 180°, we can find the measure of angle C by subtracting the measures of angle A and angle B from 180°:
[tex]m\angle C = 180^{\circ}-m\angle A - m\angle B\\\\m\angle C = 180^{\circ} - 35^{\circ} - 110^{\circ}\\\\m\angle C = 35^{\circ}[/tex]
Therefore, angles A and C both measure 35°.
