Tangent Intuition Math Example 2

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

Example 2

hard
From external point Q(7,0)Q(7, 0), find the length of the tangent to the circle x2+y2=25x^2 + y^2 = 25.

Solution

  1. 1
    Step 1: Let TT be the point of tangency. The radius OT=5OT = 5 and the tangent QTQT is perpendicular to OTOT, forming a right triangle OTQOTQ.
  2. 2
    Step 2: OQ=72+02=7OQ = \sqrt{7^2 + 0^2} = 7.
  3. 3
    Step 3: By the Pythagorean theorem: QT2+OT2=OQ2⇒QT2=49−25=24⇒QT=24=26QT^2 + OT^2 = OQ^2 \Rightarrow QT^2 = 49 - 25 = 24 \Rightarrow QT = \sqrt{24} = 2\sqrt{6}.

Answer

Tangent length =26= 2\sqrt{6}.
The tangent from an external point to a circle is perpendicular to the radius at the point of tangency. This right angle creates a right triangle with hypotenuse equal to the distance from the external point to the centre, allowing the Pythagorean theorem to find the tangent length.

About Tangent Intuition

A line that just barely touches a curve at exactly one point without crossing it, matching the curve's direction at that point.

Learn more about Tangent Intuition →

More Tangent Intuition Examples

Example 1 medium

Find the equation of the tangent line to the circle [formula] at point [formula].

Example 3 medium

Verify that the line [formula] is tangent to the circle [formula], and find the point of tangency.

Example 4 easy

At what angle does the tangent to a circle meet the radius drawn to the point of tangency? Use this

← All Tangent Intuition Examples Tangent Intuition Concept