Question

1. if z is the circum - center of △stu, sy = 19, tz = 21, and st = 30, find each measure.

sz =
yu =
wt =
zy =

Explanation:

Step1: Recall property of circum - center

The circum - center of a triangle is equidistant from the vertices of the triangle. Since \(Z\) is the circum - center of \(\triangle STU\), \(SZ = TZ\). Given \(TZ = 21\), so \(SZ=21\).

Step2: Recall property of perpendicular bisector

The circum - center is the intersection of the perpendicular bisectors of the sides of the triangle. If \(ZY\) is part of the perpendicular bisector of \(SU\), then \(SY = YU\). Given \(SY = 19\), so \(YU = 19\).

Step3: Recall property of perpendicular bisector

If \(ZW\) is part of the perpendicular bisector of \(ST\), then \(WT=\frac{ST}{2}\). Given \(ST = 30\), so \(WT=\frac{30}{2}=15\).

Step4: Use Pythagorean theorem in right - triangle \(SZY\)

In right - triangle \(SZY\), \(SZ = 21\) and \(SY = 19\). By the Pythagorean theorem \(ZY=\sqrt{SZ^{2}-SY^{2}}=\sqrt{21^{2}-19^{2}}=\sqrt{(21 + 19)(21 - 19)}=\sqrt{40\times2}=\sqrt{80}=4\sqrt{5}\)

Answer:

\(SZ = 21\)
\(YU = 19\)
\(WT = 15\)
\(ZY = 4\sqrt{5}\)