Question

b)
what point of concurrency is shown?
if op=-5x+2 and mp=2x+30, find np.

Explanation:

Step1: Identify the point of concurrency

The diagram shows perpendicular segments from point \( P \) to the sides of the triangle, so \( P \) is the incenter (point of concurrency of angle bisectors, and also equidistant from all sides). Thus, \( OP = MP = NP \) (distances from incenter to sides are equal).

Step2: Set \( OP = MP \) to solve for \( x \)

Given \( OP = -5x + 2 \) and \( MP = 2x + 30 \), set them equal:
\( -5x + 2 = 2x + 30 \)

Step3: Solve for \( x \)

Subtract \( 2x \) from both sides: \( -7x + 2 = 30 \)
Subtract 2 from both sides: \( -7x = 28 \)
Divide by -7: \( x = -4 \)

Step4: Find \( OP \) (or \( MP \), then \( NP \))

Substitute \( x = -4 \) into \( OP \):
\( OP = -5(-4) + 2 = 20 + 2 = 22 \)
Since \( NP = OP \) (incenter property), \( NP = 22 \).

Answer:

The point of concurrency is the incenter, and \( NP = 22 \).