Question

triangle abc is shown below with all three side lengths known.
let’s find the measure of \\( \angle b \\). fill in the missing parts in each step.
\\( b^2 = a^2 + c^2 - 2ac \cdot \cos(m\angle b) \\)
\\( (17.6)^2 = (14)^2 + (8.5)^2 - 2(14)(8.5) \cdot \cos(m\angle b) \\)
\\( \boxed{} = \boxed{} - 238 \cdot \cos(m\angle b) \\)
\\( \boxed{} = - 238 \cdot \cos(m\angle b) \\)
\\( \boxed{} = \cos(m\angle b) \\)
therefore:
\\( m\angle b = \boxed{100.1} \\) degrees
from here we can use the law of cosines again or the law of sines to determine \\( m\angle a \\) or \\( m\angle c \\). we omit this process here since you have practiced this several times already in this investigation. however, you should think about how you would do this since you will be asked to find all missing information for triangles in the next several questions.

Explanation:

Step1: Calculate \(a^2 + c^2\)

Given \(a = 14\), \(c = 8.5\), so \(a^2=14^2 = 196\), \(c^2 = 8.5^2=72.25\). Then \(a^2 + c^2=196 + 72.25=268.25\).

Step2: Calculate \(b^2\)

Given \(b = 17.6\), so \(b^2=(17.6)^2 = 309.76\).

Step3: Substitute into Law of Cosines

From \(b^2=a^2 + c^2-2ac\cos(m\angle B)\), substitute values: \(309.76=268.25-238\cos(m\angle B)\) (since \(2ac = 2\times14\times8.5 = 238\)).

Step4: Solve for \(\cos(m\angle B)\)

Rearrange the equation: \(309.76 - 268.25=-238\cos(m\angle B)\), so \(41.51=-238\cos(m\angle B)\). Then \(\cos(m\angle B)=\frac{41.51 - 309.76}{-238}=\frac{-268.25}{-238}\)? Wait, no, correct rearrangement: \(309.76=268.25 - 238\cos(m\angle B)\) → \(309.76 - 268.25=-238\cos(m\angle B)\) → \(41.51=-238\cos(m\angle B)\) → \(\cos(m\angle B)=\frac{268.25 - 309.76}{238}=\frac{-41.51}{238}\approx - 0.1744\).

Step5: Find \(m\angle B\)

Take arccosine: \(m\angle B=\arccos(-0.1744)\approx100.1^\circ\).

Answer:

The measure of \(\angle B\) is \(100.1\) degrees.