Question

point y is the circumcenter of δdef. find fy. diagram of δdef with circumcenter y, segments fy labeled 3x + 7, ey labeled 5x - 3, and multiple choice options: 5, 11, 17, 22

Explanation:

Step1: Recall circumcenter property

The circumcenter of a triangle is equidistant from all three vertices. So, \( FY = EY \).
Given \( FY = 3x + 7 \) and \( EY = 5x - 3 \), we set them equal: \( 3x + 7 = 5x - 3 \).

Step2: Solve for x

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

Step3: 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 to all vertices, so \( FY = DY = EY \)? Wait, no, in the diagram, \( FY \) and \( EY \) are both from circumcenter to vertices? Wait, no, maybe \( FY = DY \)? Wait, no, let's check again. Wait, the problem: Point Y is circumcenter, so \( FY = EY = DY \). Wait, the expressions: \( FY = 3x + 7 \), \( EY = 5x - 3 \). So set \( 3x + 7 = 5x - 3 \). Solve: \( 7 + 3 = 5x - 3x \) → \( 10 = 2x \) → \( x = 5 \). Then \( FY = 3(5) + 7 = 15 + 7 = 22 \)? But wait, the options include 22? Wait, no, wait, maybe I made a mistake. Wait, no, let's check the diagram again. Wait, maybe \( FY = DY \)? Wait, no, the labels: L is midpoint of FD, M midpoint of ED, N midpoint of EF. So Y is circumcenter, so FY = EY = DY. So if FY is 3x+7 and EY is 5x-3, then 3x+7=5x-3 → x=5. Then FY=35+7=22. But wait, the options have 22 as an option. Wait, but let me check again. Wait, maybe the segments: N is midpoint of EF, so EN=NF. L is midpoint of FD, so FL=LD. M is midpoint of ED, so EM=MD. So Y is circumcenter, so FY=EY=DY. So yes, set 3x+7=5x-3, solve x=5, then FY=35+7=22. Wait, but the options are 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 according to the calculation, FY=22. Wait, but let me re-express:

Wait, step1: Circumcenter is equidistant from vertices, so \( FY = EY \).
Step2: \( 3x + 7 = 5x - 3 \) → \( 2x = 10 \) → \( x = 5 \).
Step3: \( FY = 3(5) + 7 = 22 \).

Answer:

22