Question

mp is tangent to ⊙c at m, and np is tangent to ⊙c at n. find the value of x.

Explanation:

Step1: Recall Tangent Segment Theorem

Tangent segments from a common external point to a circle are congruent. So, \( MP = NP \).

Step2: Set Up Equation

Given \( NP = 7 \) and \( MP = -8x + 55 \), we set them equal: \( -8x + 55 = 7 \).

Step3: Solve for \( x \)

Subtract 55 from both sides: \( -8x = 7 - 55 \) → \( -8x = -48 \).
Divide both sides by -8: \( x = \frac{-48}{-8} = 6 \).

Answer:

\( x = 6 \)