Question

pu = 1 and xp = 3. find xv.

Explanation:

Step1: Use the property of the in - center

Since \(P\) is the in - center of \(\triangle XYZ\) (the point of intersection of angle bisectors and the center of the inscribed circle), \(PU = PV=PW\) (the distances from the in - center to the sides of the triangle are equal).
Given \(PU = 1\), so \(PV=1\).

Step2: Use the Pythagorean theorem in \(\triangle XPV\)

In right - triangle \(XPV\), by the Pythagorean theorem \(XV=\sqrt{XP^{2}+PV^{2}}\).
We know that \(XP = 3\) and \(PV = 1\).
Substitute the values into the formula: \(XV=\sqrt{3^{2}+1^{2}}=\sqrt{9 + 1}=\sqrt{10}\).

Answer:

\(\sqrt{10}\)