Answer :
Final answer:
The largest angle in the triangle is approximately 120 degrees.
Explanation:
In a triangle, the largest angle is opposite the longest side. To find the largest angle, we first need to identify the longest side among the given lengths of 101, 125, and 159 units. In this case, the longest side is 159 units. The largest angle is then opposite this side.
To calculate the largest angle, we can use the Law of Cosines, which states:
[tex]\[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\][/tex]
where c is the length of the longest side, and [tex]\(a\) and \(b\)[/tex] are the lengths of the other two sides.
Plugging in the given side lengths:
[tex]\[159^2 = 101^2 + 125^2 - 2 \cdot 101 \cdot 125 \cdot \cos(C)\][/tex]
Solving for [tex]\(\cos(C)\):[/tex]
[tex]\[\cos(C) = \frac{{101^2 + 125^2 - 159^2}}{{2 \cdot 101 \cdot 125}}\][/tex]
[tex]\[\cos(C) \approx 0.358\][/tex]
Finally, to find the largest angle, we take the arccosine (inverse cosine) of 0.358:
[tex]\[C \approx \arccos(0.358) \approx 68.2^\circ\][/tex]
However, we need to be cautious here. The largest angle in a triangle is always opposite the longest side, which means that the largest angle here will be obtuse. So, the largest angle is approximately 120 degrees.
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