Question

if p is the in - center of △jkl, find each measure.
3x + 14
9x - 34
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, the lengths from the in - center to the sides are equal. Let's assume that the segments from the in - center \(P\) to the sides are equal, say \(PN = PO\). Then we set up the equation \(3x + 14=9x - 34\).

Step2: Solve the equation for \(x\)

Subtract \(3x\) from both sides: \(14 = 9x-3x - 34\), which simplifies to \(14 = 6x-34\). Then add 34 to both sides: \(14 + 34=6x\), so \(48 = 6x\). Divide both sides by 6: \(x=\frac{48}{6}=8\).

Step3: Find \(PN\)

Substitute \(x = 8\) into the expression for \(PN\) (since \(PN=3x + 14\)). Then \(PN=3\times8 + 14=24 + 14=38\).

Step4: Find \(PO\)

Since \(PO = 9x-34\), substitute \(x = 8\). So \(PO=9\times8-34=72 - 34 = 38\).

Step5: Find \(PM\)

Because the in - center is equidistant from the sides of the triangle, \(PM=PN = PO=38\).

Answer:

\(PN = 38\)
\(PO = 38\)
\(PM = 38\)