Question

point y is the circumcenter of \\( \triangle def \\). find fy. 5 11 17 22

Explanation:

Step1: Recall circumcenter property

The circumcenter of a triangle is equidistant from all three vertices, so \( FY = EY \).

Step2: Set up equation

Given \( FY = 3x + 7 \) and \( EY = 5x - 3 \), set them equal: \( 3x + 7 = 5x - 3 \).

Step3: Solve for \( x \)

Subtract \( 3x \) from both sides: \( 7 = 2x - 3 \).
Add 3 to both sides: \( 10 = 2x \).
Divide by 2: \( x = 5 \).

Step4: Find \( FY \)

Substitute \( x = 5 \) into \( FY = 3x + 7 \): \( FY = 3(5) + 7 = 15 + 7 = 22 \)? Wait, no, wait—wait, maybe I mixed up. Wait, no, wait, the circumcenter is equidistant from vertices, but maybe \( FY = DY = EY \)? Wait, no, let's check again. Wait, the segments: \( FY = 3x + 7 \), \( EY = 5x - 3 \). Wait, but if \( Y \) is circumcenter, then \( FY = EY \), so solving \( 3x + 7 = 5x - 3 \):
\( 7 + 3 = 5x - 3x \)
\( 10 = 2x \)
\( x = 5 \). Then \( FY = 3(5) + 7 = 22 \)? But wait, the options have 22, but let me check again. Wait, maybe I made a mistake. Wait, no, the diagram: \( FN \) and \( EN \) are equal (marks), so \( YN \) is perpendicular bisector? Wait, no, \( Y \) is circumcenter, so it's the intersection of perpendicular bisectors. So \( FY = EY \), so equation is correct. Then \( x = 5 \), \( FY = 3*5 +7 = 22 \). Wait, but let me check the options: 5,11,17,22. So 22 is an option. Wait, but maybe I messed up the segments. Wait, no, the problem says "Find FY". So after solving, \( FY = 22 \)? Wait, no, wait, maybe \( FY = EY \), but let's re-express. Wait, \( 3x +7 = 5x -3 \) → \( 2x = 10 \) → \( x=5 \). Then \( FY = 3*5 +7 = 22 \). So the answer should be 22? Wait, but let me check again. Wait, maybe the segments are \( FY = 3x +7 \) and \( EY = 5x -3 \), so equate them. Yes, that's correct for circumcenter (equidistant from vertices). So \( x=5 \), \( FY=22 \).

Answer:

22