Question

1. $\triangle mnp$ shown below is not drawn to scale. as shown, $mn = 12$ and $pn = 3$. answer the following questions:\

(a) give an example of a length of $\overline{mp}$ that would not be possible. justify.\
(b) if the length of $\overline{mp}$ is an integer, list all possible values for\
$mp$. show how you arrived at your answer.

Explanation:

Step1: Apply the triangle inequality theorem

The triangle inequality theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), the following must hold: \(a + b>c\), \(a + c>b\), and \(b + c>a\). In \(\triangle MNP\) with \(MN = 12\) and \(PN = 3\), let \(MP=x\). Then \(x+3>12\) (i.e., \(x > 9\)) and \(x+12>3\) (always true for positive \(x\)) and \(3 + 12>x\) (i.e., \(x<15\)).

Step2: Find a non - possible length for part (a)

Let's choose \(x = 5\). Since \(5+3=8<12\), by the triangle inequality theorem (\(MP + PN>MN\) must hold), a length of \(5\) for \(\overline{MP}\) is not possible.

Step3: List integer values for part (b)

From \(9 < x<15\) (where \(x = MP\) is an integer), the possible integer values of \(x\) are \(10\), \(11\), \(12\), \(13\), \(14\)

Answer:

(a) An example of a non - possible length for \(\overline{MP}\) is \(5\). Justification: By the triangle inequality \(MP+PN>MN\), and \(5 + 3=8<12\).

(b) The possible integer values for \(MP\) are \(10\), \(11\), \(12\), \(13\), \(14\)