Question

m n 24 127 29 p claim i: mp > mn claim ii: np < mn whose claim is true? a) only claim i is true. b) only claim ii is true. c) both claims are true. d) neither of them is true.

Explanation:

Step1: Find the third angle in triangle MNP

In a triangle, the sum of interior angles is \(180^\circ\). Given \(\angle MNP = 127^\circ\) and \(\angle MPN=29^\circ\), let \(\angle NMP = x\). Then \(x + 127^\circ+ 29^\circ=180^\circ\), so \(x=180^\circ-(127^\circ + 29^\circ)=24^\circ\).

Step2: Apply the Law of Sines

The Law of Sines states that in a triangle \(\frac{MN}{\sin\angle MPN}=\frac{NP}{\sin\angle NMP}=\frac{MP}{\sin\angle MNP}\). Let \(MN = a\), \(NP = b\), \(MP = c\). We know \(\sin127^\circ=\sin(180^\circ - 53^\circ)=\sin53^\circ\approx0.8\), \(\sin29^\circ\approx0.48\), \(\sin24^\circ\approx0.41\).
From \(\frac{MN}{\sin29^\circ}=\frac{MP}{\sin127^\circ}\), so \(MP=\frac{MN\times\sin127^\circ}{\sin29^\circ}\approx\frac{MN\times0.8}{0.48}=\frac{5}{3}MN\approx1.67MN\), so \(MP > MN\) (Claim I is true).
From \(\frac{MN}{\sin29^\circ}=\frac{NP}{\sin24^\circ}\), so \(NP=\frac{MN\times\sin24^\circ}{\sin29^\circ}\approx\frac{MN\times0.41}{0.48}\approx0.85MN\), so \(NP < MN\) (Claim II is true).

Answer:

c) Both claims are true.