Question

if \\( \overline { p q } \\) and \\( \overline { p r } \\) are tangent to \\( \odot e \\) in the given figure, find the value of \\( x \\).

\\( x = \square \\) (simplify your answer.)

Explanation:

Step1: Use the tangent - segment theorem

If two tangent segments are drawn to a circle from the same external point, then the two tangent segments are congruent. So, \(PR = PQ\).
Given \(PR=7x - 15\) and \(PQ = 3x+5\), we set up the equation \(7x-15=3x + 5\).

Step2: Solve the linear equation for \(x\)

Subtract \(3x\) from both sides of the equation:
\(7x-3x-15=3x-3x + 5\)
\(4x-15=5\)
Add \(15\) to both sides:
\(4x-15 + 15=5+15\)
\(4x=20\)
Divide both sides by \(4\):
\(x=\frac{20}{4}\)

Answer:

\(x = 5\)