Question

1. point p is the circumcenter of △xyz. use the given information to find pz. px = 3x + 2; py = 4x - 8; (with an image of triangle xyz with right angles at y, a, b and point p inside) 16. point p is the circumcenter of △xyz. use the given information to find py. px = 4x + 3; pz = 6x - 11; (with an image of triangle xyz with right angles at c, d, e and point p inside)

Explanation:

Problem 15

Step1: Use the property of circumcenter

Since \(P\) is the circumcenter of \(\triangle XYZ\), \(PX = PY\).
So, \(3x + 2=4x - 8\).

Step2: Solve the equation for \(x\)

Subtract \(3x\) from both sides: \(3x+2 - 3x=4x - 8-3x\), which gives \(2=x - 8\).
Add \(8\) to both sides: \(2 + 8=x-8 + 8\), so \(x = 10\).

Step3: Find \(PZ\)

Since \(PX=PZ\) (circum - center property), and \(PX=3x + 2\). Substitute \(x = 10\) into \(PX\): \(PX=3\times10+2=32\). So \(PZ = 32\).

Step1: Use the property of circumcenter

Since \(P\) is the circumcenter of \(\triangle XYZ\), \(PX = PZ\).
So, \(4x + 3=6x-11\).

Step2: Solve the equation for \(x\)

Subtract \(4x\) from both sides: \(4x + 3-4x=6x-11 - 4x\), which gives \(3 = 2x-11\).
Add \(11\) to both sides: \(3+11=2x-11 + 11\), so \(14 = 2x\).
Divide both sides by \(2\): \(x=\frac{14}{2}=7\).

Step3: Find \(PY\)

Since \(PY = PX\) (circum - center property), and \(PX=4x + 3\). Substitute \(x = 7\) into \(PX\): \(PX=4\times7+3=28 + 3=31\). So \(PY = 31\).

Answer:

\(32\)

Problem 16