Question

consider triangle lmn, where p is the circumcenter of triangle lmn, mp = (x + 12) units, and lp = (4x + 3) units. what is the length of pm, in units?

Explanation:

Step1: Recall Circumcenter Property

The circumcenter \( P \) of a triangle is equidistant from all vertices, so \( LP = MP = NP \). Thus, set \( 4x + 3 = x + 12 \).

Step2: Solve for \( x \)

Subtract \( x \) from both sides: \( 3x + 3 = 12 \).
Subtract 3: \( 3x = 9 \).
Divide by 3: \( x = 3 \).

Step3: Find \( PM \)

Substitute \( x = 3 \) into \( MP = x + 12 \): \( PM = 3 + 12 = 15 \).

Answer:

15