Question

if p is the in - center of △jkl, find each measure.
k
3x + 14
m
p
n
j
o 9x - 34
l
x = 38
pn = type your answer..
po = type your answer..
pm = type your answer..

Explanation:

Step1: Recall in - center property

The in - center of a triangle is equidistant from the sides of the triangle. So \(PN = PM=PO\).

Step2: Find \(PN\)

We know that \(PN = 3x + 14\) and \(x = 38\). Substitute \(x = 38\) into the expression for \(PN\).
\[PN=3\times38 + 14\]
\[PN = 114+14\]
\[PN = 128\]

Step3: Find \(PO\)

Since \(PO=9x - 34\) and \(x = 38\), substitute \(x = 38\) into the expression for \(PO\).
\[PO=9\times38-34\]
\[PO = 342-34\]
\[PO = 308\] (This is wrong. Since \(PN = PM = PO\) because \(P\) is the in - center, we should use the correct equal - distance relationship. Let's start over for \(PO\))
Since \(P\) is the in - center, \(PO=PN\). Substitute \(x = 38\) into \(3x + 14\) (the expression for \(PN\)). So \(PO=3\times38 + 14=128\)

Step4: Find \(PM\)

Since \(P\) is the in - center, \(PM = PN\). Substitute \(x = 38\) into \(3x + 14\). So \(PM=3\times38+14 = 128\)

Answer:

\(PN = 128\)
\(PO = 128\)
\(PM = 128\)