Question

the side lengths of triangle abc are written in terms of the variable p, where p ≥ 3. diagram of triangle abc with vertices a, b, c; side ac: p + 4, side ab: 4p − 1, side bc: 3p which is correct regarding the angles of the triangle?
○ ( mangle a > mangle c > mangle b )
○ ( mangle b > mangle a > mangle c )
○ ( mangle c > mangle a > mangle b )
○ ( mangle c > mangle b > mangle a )

Explanation:

Step1: Compare side lengths

To determine the angles, we first compare the side lengths. The sides are \( AB = 4p - 1 \), \( BC = 3p \), and \( AC = p + 4 \) with \( p \geq 3 \).

Let's find the order of the sides. For \( p \geq 3 \):

• Compare \( 4p - 1 \) and \( 3p \): \( 4p - 1 - 3p = p - 1 \). Since \( p \geq 3 \), \( p - 1 \geq 2 > 0 \), so \( 4p - 1 > 3p \).
• Compare \( 3p \) and \( p + 4 \): \( 3p - (p + 4) = 2p - 4 \). For \( p \geq 3 \), when \( p = 3 \), \( 2(3) - 4 = 2 > 0 \), so \( 3p > p + 4 \) for \( p \geq 3 \).

Thus, the order of the sides is \( AB > BC > AC \) (since \( 4p - 1 > 3p > p + 4 \) for \( p \geq 3 \)).

Step2: Relate sides to angles

In a triangle, the larger side is opposite the larger angle.

• Side \( AB \) is opposite \( \angle C \), so \( m\angle C \) is opposite \( AB \).
• Side \( BC \) is opposite \( \angle A \), so \( m\angle A \) is opposite \( BC \).
• Side \( AC \) is opposite \( \angle B \), so \( m\angle B \) is opposite \( AC \).

Since \( AB > BC > AC \), the angles opposite them follow the same order: \( m\angle C > m\angle A > m\angle B \).

Answer:

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