Point A is the point of concurrency of the angle bisectors of ΔDEF. Point A is the point of concurrency of triangle D E F. Lines are drawn from each point of the triangle to point A. Lines are drawn from point A to the sides of the triangle to form right angles and line segments A X, A Y, and A Z. The length of F A is 6 centimeters, the length of A D is 5 centimeters, the length of A X is 3 centimeters, and the length of Y D is 4 centimeters. What is the length of ZA? A) ZA = 3cm B) ZA = 4cm C) ZA = 5cm D) ZA = 6cm

Answer:A) ZA = 3 cmStep-by-step explanation:The triangle is shown below.From the triangle, A is the concurrency point of angle bisectors of all vertices.Consider ΔAYD,Using Pythagorean theorem,[tex]AD^2=AY^2+ YD^2\\5^2=AY^2+4^2\\25=AY^2+16\\AY^2=25-16\\AY=\sqrt{9}=3[/tex]Consider triangles ADY and ADZ.[tex]\angle AYD\cong \angle AZD=90\\\angle ADY\cong \angle ADZ \textrm{ (angle bisector) }\\AD\cong AD\textrm{ (Common side)}[/tex]The two triangle are congruent by AAS postulate.Therefore, by CPCTE, [tex]AY=ZA=3\textrm{ cm}[/tex]