Question

use \\( \triangle a b c \\) shown below. which side is the longest?

choose the correct answer below.

\\( \bigcirc \\) a. \\( \overline{a b} \\)
\\( \bigcirc \\) b. \\( \overline{b c} \\)
\\( \bigcirc \\) c. \\( \overline{a c} \\)

Explanation:

Step1: Calculate angle at A

The angle at \( A\) inside the triangle is \(180^{\circ}- 75^{\circ}=105^{\circ}\) (supplementary angles).

Step2: Use triangle angle - side relationship

In a triangle, the larger the angle, the longer the side opposite it.
The sum of angles in \(\triangle ABC\) is \(180^{\circ}\). Let \(\angle B = \theta\). Then \(\theta=180^{\circ}-105^{\circ}-\angle C\).
We can also use the Law of Cosines: \(BC^{2}=AB^{2}+AC^{2}-2(AB)(AC)\cos A\)
Substitute \(AB = 4\), \(AC = 5\), \(A = 105^{\circ}\), \(\cos105^{\circ}=\cos(60^{\circ} + 45^{\circ})=\cos60^{\circ}\cos45^{\circ}-\sin60^{\circ}\sin45^{\circ}=\frac{\sqrt{2}-\sqrt{6}}{4}\approx - 0.2588\)
\(BC^{2}=4^{2}+5^{2}-2\times4\times5\times(- 0.2588)=16 + 25+10.352=51.352\)
\(AB^{2}=16\), \(AC^{2}=25\)

Since \(BC^{2}>AC^{2}>AB^{2}\), then \(BC>AC>AB\)

Answer:

B. \(\overline{BC}\)