Student can prove and apply Theorem 10.1 (tangent is perpendicular to radius at point of contact) to find missing lengths.
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Figure 10.5 showing radius OP and point Q on tangent XY.
Clean geometric diagram showing a circle with center O, a tangent line XY touching the circle at point P, a solid radius…
Statement and proof of the tangent-radius theorem.
Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
This fundamental property connects circles and straight lines, allowing us to form right triangles and solve for missing distances.
Formalizes the perpendicularity as a tool to use the Pythagorean theorem.
When a line is tangent to a circle, it forms a right angle with the radius at the point of contact. This creates a right triangle when connected to the center.
Worked example based on Exercise 10.1 Q3.
A tangent at a point of a circle of radius 5 cm meets a line through the centre at a point so that cm. Find Length .
Given:
Faded calculation finding the radius given tangent and hypotenuse.
Let's practice finding the radius. From a point Q, the length of the tangent to a circle is 24 cm () and the distance of Q from the centre is 25 cm (). Because the radius is perpendicular to the tangent, is a right triangle. Using the Pythagorean theorem, we set up the equation . Substituting the numerical values gives us . Solving this equation yields , which means the radius cm.
MCQ testing tangent perpendicularity application.
The length of a tangent from a point A at a distance of 5 cm from the centre of the circle is 4 cm. What is the radius of the circle?