Question

solve for x. assume that lines which appear to be tangent are tangent.
8)
diagram of a circle with two tangent segments labeled -7 + 2x and x + 1, and a problem 8 box
9)
diagram of a circle with two tangent segments labeled 5x + 2 and 11 + 4x, and a problem 9 box

Explanation:

Problem 8:

Step1: Recall tangent segment theorem

Tangent segments from a common external point to a circle are equal in length. So, we set the two tangent expressions equal: $-7 + 2x = x + 1$.

Step2: Solve for x

Subtract $x$ from both sides: $-7 + 2x - x = x + 1 - x$ → $-7 + x = 1$. Then add 7 to both sides: $x = 1 + 7$ → $x = 8$.

Step1: Apply tangent segment theorem

Tangent segments from a common external point to a circle are congruent. Thus, $5x + 2 = 11 + 4x$.

Step2: Solve for x

Subtract $4x$ from both sides: $5x + 2 - 4x = 11 + 4x - 4x$ → $x + 2 = 11$. Then subtract 2 from both sides: $x = 11 - 2$ → $x = 9$.

Answer:

$x = 8$

Problem 9: