Question

the side lengths of triangle abc are written in terms of the variable p, where p ≥ 3. which is correct regarding the angles of the triangle? m∠a > m∠c > m∠b m∠b > m∠a > m∠c m∠c > m∠a > m∠b m∠c > m∠b > m∠a

Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the larger angle is opposite the longer side, and the smaller angle is opposite the shorter side. So we need to compare the lengths of the sides of triangle \(ABC\) to determine the order of the angles. The sides are \(AB = 4p - 1\), \(BC=3p\), and \(AC = p + 4\), with \(p\geq3\).

Step2: Compare \(AB\) and \(BC\)

We want to find when \(4p-1>3p\). Subtract \(3p\) from both sides: \(4p - 1-3p>3p - 3p\), which simplifies to \(p - 1>0\), or \(p > 1\). Since \(p\geq3\), \(4p - 1>3p\) for all \(p\geq3\). So \(AB>BC\).

Step3: Compare \(BC\) and \(AC\)

We want to find when \(3p>p + 4\). Subtract \(p\) from both sides: \(3p-p>p + 4-p\), which gives \(2p>4\), or \(p > 2\). Since \(p\geq3\), \(3p>p + 4\) for all \(p\geq3\). So \(BC>AC\).

Step4: Determine the side - length order

From the above comparisons, we have \(AB = 4p-1\), \(BC = 3p\), \(AC=p + 4\). Since \(4p-1>3p\) and \(3p>p + 4\) (for \(p\geq3\)), the order of the side lengths from longest to shortest is \(AB>BC>AC\).

Step5: Determine the angle order

The angle opposite \(AB\) is \(\angle C\), the angle opposite \(BC\) is \(\angle A\), and the angle opposite \(AC\) is \(\angle B\). Since in a triangle, the larger the side, the larger the opposite angle, we have \(m\angle C>m\angle A>m\angle B\) (because \(AB\) (opposite \(\angle C\)) is the longest, \(BC\) (opposite \(\angle A\)) is the second - longest, and \(AC\) (opposite \(\angle B\)) is the shortest).

Answer:

\(m\angle C>m\angle A>m\angle B\) (the option: \(m\angle C>m\angle A>m\angle B\))