Grade 10

Advanced

7 min

Lesson 4 of 12

Circle Geometry: Chords and Tangents

Not Started

Investigates the relationship between circles, their chords, and tangent lines in complex diagrams.

Did you know that every time a car wheel touches the road, it creates a perfect right angle that engineers use to calculate braking distance? This hidden geometry is what keeps vehicles stable on every curve you drive.

Learning Objectives

1.Prove and apply the theorem that a tangent line is perpendicular to the radius at the point of tangency.
2.Calculate unknown lengths using the Perpendicular Chord Bisector Theorem.
3.Solve algebraic problems involving equal tangent segments from a common external point.

The Perpendicular Chord Bisector

A chord is a straight line segment whose endpoints both lie on a circle. One of the most powerful tools in circle geometry is the Perpendicular Chord Bisector Theorem. It states that if a radius (or diameter) of a circle is perpendicular to a chord, then it bisects the chord (cuts it into two equal halves). Conversely, the perpendicular bisector of any chord must pass through the center of the circle. This creates right-angled triangles within the circle, allowing us to use the Pythagorean Theorem:

a^2 + b^2 = c^2

where

c

is always the radius of the circle.

A circle has a radius of $10$ cm. A chord is drawn $6$ cm from the center. How long is the chord?

1. Identify the right triangle formed by the radius ( $r=10$ ), the distance from the center ( $d=6$ ), and half the chord ( $x$ ). 2. Apply Pythagoras: $6^2 + x^2 = 10^2$ . 3. Solve for $x$ : $36 + x^2 = 100 \\rightarrow x^2 = 64 \\rightarrow x = 8$ . 4. Since the radius bisects the chord, the total length is $2x = 2(8) = 16$ cm.

Quick Check

If a chord is $24$ units long and is bisected by a perpendicular radius, what is the length of each segment of the chord?

Bisection means dividing into two equal parts.

Answer

12 units.

The Tangent-Radius Relationship

A tangent is a line that touches a circle at exactly one point, known as the point of tangency. The fundamental rule of tangents is that a tangent is always perpendicular ( $\\perp$ ) to the radius at that point. This means the angle between the radius and the tangent line is always $90^\circ$ . This relationship allows us to connect the circle's internal properties to external points, often forming right triangles with the center of the circle.

Point $P$ is outside a circle with center $O$ . The distance $OP$ is $13$ cm. If the radius of the circle is $5$ cm, find the length of the tangent segment $PT$ from point $P$ to the point of tangency $T$ .

1. Recognize that $\\triangle OTP$ is a right triangle with the right angle at $T$ . 2. The hypotenuse is $OP = 13$ and one leg is the radius $OT = 5$ . 3. Use Pythagoras: $5^2 + PT^2 = 13^2$ . 4. $25 + PT^2 = 169 \\rightarrow PT^2 = 144$ . 5. $PT = \\sqrt{144} = 12$ cm.

Quick Check

What is the measure of the angle formed by a radius and a tangent line at the point of contact?

Answer

90 degrees (or a right angle).

Tangents from a Common External Point

When two tangent segments are drawn to a circle from the same external point, they are always equal in length. This is known as the Tangents from a Common Point Theorem. If you draw tangents from point $P$ to points $A$ and $B$ on a circle, then $PA = PB$ . This symmetry often creates an isosceles triangle or a kite shape when connected to the circle's center, which is extremely useful for solving complex geometric proofs.

Two tangents are drawn from point $P$ to a circle at points $A$ and $B$ . If $PA = 3x + 15$ and $PB = 5x - 7$ , solve for $x$ and find the length of the tangents.

1. Set the expressions equal to each other: $3x + 15 = 5x - 7$ . 2. Subtract $3x$ from both sides: $15 = 2x - 7$ . 3. Add $7$ to both sides: $22 = 2x$ . 4. Divide by $2$ : $x = 11$ . 5. Substitute $x$ back to find the length: $3(11) + 15 = 33 + 15 = 48$ units.

Key Takeaways

1A radius perpendicular to a chord bisects that chord.
2The angle between a radius and a tangent at the point of tangency is always $90^\circ$ .
3Tangent segments from the same external point to a circle are congruent (equal).
4The Pythagorean Theorem ( $a^2 + b^2 = c^2$ ) is the primary tool for calculating lengths in these diagrams.

Quick Check

Multiple Choice

A chord of length $10$ is $12$ units from the center of a circle. What is the radius of the circle?

True/False

If two lines are tangent to the same circle from the same external point, the triangle formed by the two points of tangency and the external point is always isosceles.

Multiple Choice

In a circle with center $O$ , radius $OA=7$ meets tangent $AP$ at point $A$ . If $AP=24$ , what is the distance $OP$ ?

What's Next?

Review Tomorrow

In 24 hours, try to sketch a circle with two tangents from an external point and explain to yourself why the resulting shape is a kite.

Practice Activity

Find a circular object at home (like a clock or plate) and calculate where a $10$ cm chord would sit if the radius is known.

Browse

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Circle Geometry: Chords and Tangents

Learning Objectives

The Perpendicular Chord Bisector

The Tangent-Radius Relationship

Tangents from a Common External Point

Key Takeaways

Quick Check

What's Next?

Circle Geometry: Chords and Tangents

Learning Objectives

The Perpendicular Chord Bisector

The Tangent-Radius Relationship

Tangents from a Common External Point

Key Takeaways

Quick Check

What's Next?