Question

1. △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:

Part (a)

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. For \(\triangle MNP\) with sides \(MN = 12\), \(NP=3\), and \(MP = x\) (let's say), we have three inequalities:

1. \(MN + NP>MP\) i.e., \(12 + 3>x\) or \(x < 15\)
2. \(MN+MP>NP\) i.e., \(12 + x>3\) (this is always true for positive \(x\))
3. \(NP + MP>MN\) i.e., \(3 + x>12\) or \(x>9\)

So, \(9 < x<15\). A length that is not possible should be outside this range. Let's take \(x = 5\). Since \(5<9\), it violates the triangle inequality \(3 + 5=8
ot>12\).

Step2: Justify

For \(MP = 5\), check the triangle inequalities:

• \(MN+NP=12 + 3 = 15>5\) (satisfies)
• \(MN + MP=12+5 = 17>3\) (satisfies)
• \(NP + MP=3 + 5=8

ot>12\) (does not satisfy). So, \(MP = 5\) is not possible.

Step1: Recall Triangle Inequality Theorem

From the triangle inequality theorem for \(\triangle MNP\) with \(MN = 12\) and \(NP = 3\), we have:
\(\vert MN - NP\vertSubstitute the values: \(\vert12 - 3\vert\(9

Step2: Find integer values in the range

We need to find all integer values of \(MP\) such that \(9

Answer:

An example of a non - possible length for \(\overline{MP}\) is \(5\) (any value \(\leq9\) or \(\geq15\) would work, justification as above).

Part (b)