Question

given: ( p ) is the circumcenter of ( \triangle j k l ). ( overline{p z}, overline{p y} ), and ( overline{p x} ) are perpendicular bisectors.
( x y=14 mathrm{~cm}, y l=17 mathrm{~cm}, p z=8 mathrm{~cm}, p j=19 mathrm{~cm}, m angle y j p=35^{circ} ).
what is the measure of ( overline{k p} ) ?
what is the measure of ( overline{x j} ) ?
what is the measure of ( overline{j l} ) ?
what is the measure of ( angle j p y ) ?

Explanation:

Step1: Use property of circumcenter

Since \(P\) is the circumcenter of \(\triangle JKL\), \(PJ = PK=PL\). Given \(PJ = 19\mathrm{cm}\), so \(KP=19\mathrm{cm}\).

Step2: Use property of perpendicular bisector

Since \(\overline{PX}\) is a perpendicular bisector of \(JK\), \(XJ=XK\). Given \(KX = 14\mathrm{cm}\), so \(XJ = 14\mathrm{cm}\).

Step3: Use property of perpendicular bisector

Since \(\overline{PY}\) is a perpendicular bisector of \(JL\), \(JY=YL\). Given \(YL = 17\mathrm{cm}\), so \(JL=JY + YL=17+17 = 34\mathrm{cm}\).

Step4: Use triangle angle - sum property

In right - triangle \(JPY\), \(\angle JYP = 90^{\circ}\), \(\angle YJP=35^{\circ}\). Using the angle - sum property of a triangle \(\angle JPY=180^{\circ}-\angle JYP-\angle YJP\). So \(\angle JPY = 180^{\circ}-90^{\circ}-35^{\circ}=55^{\circ}\).

Answer:

\(\overline{KP}=19\mathrm{cm}\), \(\overline{XJ}=14\mathrm{cm}\), \(\overline{JL}=34\mathrm{cm}\), \(\angle JPY = 55^{\circ}\)