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The Tangent-Radius Theorem

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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Theorem 10.1

Statement and proof of the tangent-radius theorem.

Theorem 10.1: 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.

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Shortest Distance Proof

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…

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Right Triangles at the Tangent

Formalizes the perpendicularity as a tool to use the Pythagorean theorem.

Right Triangles at the Tangent

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.

Rule

∠ OPQ = 9 0^{\circ}

Tangent

⊥

Radius

Pythagoras

O Q^{2} = O P^{2} + P Q^{2}

OQ

is ALWAYS the hypotenuse

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Example: Calculating Tangent Length

Worked example based on Exercise 10.1 Q3.

Problem Statement

A tangent $PQ$ at a point $P$ of a circle of radius 5 cm meets a line through the centre $O$ at a point $Q$ so that $OQ = 12$ cm. Find Length $PQ$ .

Given:

Radius $OP = 5$ cm
Distance from center $OQ = 12$ cm

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Practice: Finding the Radius

Faded calculation finding the radius given tangent and hypotenuse.

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Let's practice finding the radius. From a point Q, the length of the tangent to a circle is 24 cm ( $PQ = 24$ ) and the distance of Q from the centre is 25 cm ( $OQ = 25$ ). Because the radius is perpendicular to the tangent, $\triangle OPQ$ is a right triangle. Using the Pythagorean theorem, we set up the equation $OP^2 +$ $^2 = OQ^2$ . Substituting the numerical values gives us $OP^2 + 24^2 =$ $^2$ . Solving this equation yields $OP^2 = 625 - 576$ , which means the radius $OP =$ cm.

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Check Your Understanding

MCQ testing tangent perpendicularity application.

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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?

← previousIntersecting Lines: Secants And Tangents next →Tangents From An External Point

Example: Calculating Tangent Length

Worked example based on Exercise 10.1 Q3.

Problem Statement

A tangent $PQ$ at a point $P$ of a circle of radius 5 cm meets a line through the centre $O$ at a point $Q$ so that $OQ = 12$ cm. Find Length $PQ$ .

Given:

Radius $OP = 5$ cm
Distance from center $OQ = 12$ cm

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Practice: Finding the Radius

Faded calculation finding the radius given tangent and hypotenuse.

0 of 3 blanks filled

Check Your Understanding

MCQ testing tangent perpendicularity application.

1 / 5

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?