E. Geometry and Measurement · Geometry and Measurement: Circles, Triangles, Measurement Systems, Pythagoras and Volume

Circle Properties: Chords and Tangents

E1.2 E1.5 A1About 70 minutes

Learning goals

I can name the parts of a circle: radius, diameter, chord, arc, sector, segment and tangent.
I can use the property that the perpendicular from the centre to a chord cuts the chord in half.
I can find the centre of a circle using the perpendicular bisectors of two chords.
I can use the property that a tangent is perpendicular to the radius, together with the Pythagorean theorem, to find unknown lengths.

Words to know

chord
A line segment joining two points on a circle. The longest chord is a diameter.

arc
Part of the circumference between two points. A minor arc is less than half the circle; a major arc is more.

sector
A "pizza slice" region between two radii and an arc.

segment (of a circle)
The region between a chord and its arc.

perpendicular bisector
A line that crosses a segment at its midpoint at 90°.

tangent
A line that touches a circle at exactly one point, called the point of tangency.

central angle
An angle with its vertex at the centre of the circle.

A hook: the broken plate

An archaeologist digs up a piece of a round clay plate. Only a curved edge is left. How can she know how big the whole plate was?

Circles have hidden rules. Once you know them, a small piece of a circle tells you where its centre is and how big it is. In this lesson you will learn the rules for chords and tangents.

The parts of a circle. A chord joins two points on the circle; a diameter is a chord through the centre. A tangent touches the circle at one point, and the radius to that point meets it at 90°.

Property 1: the perpendicular from the centre bisects a chord

Draw a chord AB and radii OA and OB. Because OA = OB, triangle OAB is isosceles. Drop a perpendicular from O to the chord at M. The two right triangles OMA and OMB share side OM and have equal hypotenuses, so by the Pythagorean theorem AM² = r² − OM² = MB². That means AM = MB: the chord is cut in half.

This works the other way too: the perpendicular bisector of any chord passes through the centre.

A chord 16 cm long in a circle of radius 10 cm (drawn to scale). The perpendicular OM splits the chord into two 8 cm halves and makes a right triangle with the radius.

Property 2: finding the centre

Since every chord's perpendicular bisector passes through the centre, draw two chords and their perpendicular bisectors. They cross at exactly one point: the centre. This is how you can find the centre of a plate, a round table or a tree stump.

Property 3: tangents

A tangent is perpendicular to the radius at the point of tangency. If you draw two tangents from the same outside point P, the tangent segments are equal: PA = PB. (The right triangles OAP and OBP share hypotenuse OP and have equal legs OA = OB, so their third sides match.)

Two tangents from P. Each one meets its radius at a right angle, so OAP and OBP are right triangles with the same hypotenuse OP. That is why PA = PB.

Worked example

Example 1: Distance from the centre

A circle has radius 10 cm. A chord is 16 cm long. How far is the chord from the centre?

The perpendicular from O bisects the chord, so each half is 16 ÷ 2 = 8 cm.
Right triangle: d² + 8² = 10².
d² = 100 − 64 = 36, so d = 6 cm.

6 cm

Worked example

Example 2: Length of a chord

A chord is 5 cm from the centre of a circle with radius 13 cm. How long is the chord?

Half the chord, h: h² + 5² = 13².
h² = 169 − 25 = 144, so h = 12 cm.
The whole chord is 2 × 12 = 24 cm.

24 cm

Worked example

Example 3: The broken plate

The edge of the plate piece forms an arc. Its chord is 20 cm long, and the arc rises 4 cm above the middle of the chord. Find the plate's radius.

Let the radius be r. The perpendicular bisector of the chord goes through the centre, so the centre is r − 4 cm below the middle of the arc, along that line.
Right triangle: half chord 10, distance from centre r − 4, hypotenuse r.
r² = 10² + (r − 4)² = 100 + r² − 8r + 16.
Subtract r² from both sides: 0 = 116 − 8r, so 8r = 116 and r = 14.5 cm.

The plate had a radius of 14.5 cm (diameter 29 cm).

Worked example

Example 4: Tangent length

Point P is 17 cm from the centre of a circle with radius 8 cm. How long is a tangent segment from P to the circle?

The radius meets the tangent at 90°, so OP is the hypotenuse.
PA² + 8² = 17², so PA² = 289 − 64 = 225.
PA = 15 cm. The other tangent, PB, is also 15 cm.

15 cm

Watch out!

The perpendicular must come from the centre. A random line through the chord's midpoint only reaches the centre if it is perpendicular.
In tangent problems, the right angle is at the point of tangency, so the line to the outside point (OP) is the hypotenuse.
Always halve the chord before using Pythagoras.

Hands-on activity

Find the centre of a plate

You need: a round plate or lid, paper, a pencil, a ruler, and a book or paper with a square corner (to make right angles).

Trace the plate on paper and take the plate away.
Draw any chord AB across the circle. Measure it and mark its midpoint.
Use the square corner of a book to draw a line through the midpoint at 90° to the chord. (Or fold the paper so A lands exactly on B; the fold is the perpendicular bisector.)
Repeat with a second chord in a different direction. Where the two lines cross is the centre.
Check: measure from your centre to three points on the circle. They should all be equal (the radius).

What to look for: the three distances match within a couple of millimetres. If not, check the right angles.

Just for fun

The tangent is the most polite line in geometry. It visits the circle, but only stays for one point.

Your turn

A circle has radius 5 cm. A chord is 8 cm long. How far is it from the centre?

A chord is 8 cm from the centre of a circle with radius 17 cm. How long is the chord?

What is the name of the longest chord in a circle?

P is 10 cm from the centre of a circle with radius 6 cm. How long is the tangent from P?

True or false: the perpendicular bisector of a chord always passes through the centre.

What angle does a tangent make with the radius at the point of tangency?

Two tangents are drawn from P to a circle. PA = 12 cm. How long is PB?

More practice: warm-up

What do we call a segment joining two points on a circle?

The radius is 7 cm. What is the diameter?

A chord that passes through the centre is called a ___.

How many points does a tangent share with a circle?

A chord is 18 cm long. How long is each half after a perpendicular from the centre cuts it?

More practice: on your own

Radius 25 cm, chord 48 cm. How far is the chord from the centre?

A chord 10 cm long is 12 cm from the centre. Find the radius.

A tangent segment from P is 20 cm and the radius is 15 cm. How far is P from the centre?

A round window has radius 50 cm. A straight brace across it is 80 cm long. How far is the brace from the centre?

Explain in two or three steps how to find the centre of a circle drawn on paper.

Challenge

An arc of a circular arch has a chord of 30 cm and rises 5 cm above the chord's midpoint. Find the radius.

A circle has radius 10 cm. Two parallel chords are 16 cm and 12 cm long, on opposite sides of the centre. How far apart are they? What if they were on the same side?

P is 26 cm from the centre O, and the tangent PA is 24 cm. Find the radius, then the area of the kite OAPB formed by both tangents and both radii.

Check your understanding

Quiz

A segment that joins two points on a circle is a:

A circle has radius 13 cm. A chord is 5 cm from the centre. How long is the chord?

What angle does a tangent make with the radius at the point of tangency?

How can you find the centre of a circle?

P is 5 cm from the centre of a circle with radius 3 cm. How long is the tangent from P?

Two tangents from P touch a circle at A and B. Which is always true?

For parents and teachers

Big idea: Expectation E1.2 asks students to create and analyse designs using geometric relationships and circle and triangle properties. This lesson covers chord and tangent properties, using the Pythagorean theorem from Grade 8 (and E1.5) to find lengths.

Common difficulties:

Forgetting to halve the chord before using Pythagoras.
Using the wrong hypotenuse. In a chord problem the radius is the hypotenuse; in a tangent problem the line from the centre to the outside point is the hypotenuse.
Algebra in the broken-plate problem. Expanding (r − 4)² = r² − 8r + 16 is the tricky step; the algebra unit covers this.

How to help: Do the plate-centre activity together. Ask "How do you know that's the centre?" and let your teen explain the property in their own words.