Tangent to a Circle Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Two tangent segments PAPA and PBPB are drawn from external point PP to circle OO. If PA=3x−4PA = 3x - 4 and PB=x+8PB = x + 8, find the lengths of both tangent segments.

Solution

  1. 1
    Step 1: Apply the Two-Tangent Theorem: tangent segments from the same external point to a circle are equal in length. So PA=PBPA = PB.
  2. 2
    Step 2: Set up the equation: 3x−4=x+83x - 4 = x + 8.
  3. 3
    Step 3: Solve for xx: 3x−x=8+4⇒2x=12⇒x=63x - x = 8 + 4 \Rightarrow 2x = 12 \Rightarrow x = 6.
  4. 4
    Step 4: Substitute back: PA=3(6)−4=14PA = 3(6) - 4 = 14 and PB=6+8=14PB = 6 + 8 = 14. Both equal 1414 units.

Answer

PA=PB=14PA = PB = 14 units
The Two-Tangent Theorem guarantees that both tangent segments from an external point are congruent. Setting the expressions equal and solving yields x = 6, making each segment 14 units long.

About Tangent to a Circle

A line that touches a circle at exactly one point, called the point of tangency. At this point, the tangent line is perpendicular to the radius.

Learn more about Tangent to a Circle →

More Tangent to a Circle Examples

Example 1 easy

A tangent line touches circle [formula] at point [formula]. The radius [formula] cm. A line from an

Example 3 easy

Line [formula] is tangent to circle [formula] at point [formula]. If the radius [formula] and a poin

Example 4 hard

A circle with center [formula] and radius [formula] is inscribed in angle [formula] (i.e., tangent t

← All Tangent to a Circle Examples Tangent to a Circle Concept