Chord of a circle

Here you will learn about chords of circles. You will learn what a chord of a circle is, theorems that involve chords, and the application of these theorems. You will explore the proof of the theorems and how to use them to solve more complex problems.

Students first learn how to work with these circles and chords in a geometry course where they spend time delving into the proofs of theorems and application of those theorems. Circle theorems are essential for success on graduation assessments as well as college entrance assessments.

What is the chord of a circle?

The chord of a circle is a straight line that connects two points on the edge of the circle or circumference of a circle. The longest chord in a circle is the diameter of the circle.

As the chords of a circle get further away from the center of the circle, the shorter they become.

Looking at circle A, you can visually see how chord BF is shorter than chord CE. Line segment DG has endpoints on the edge of the circle (circumference of the circle) and goes through the center of the circle. This means that chord DG is the diameter of the circle and also the longest chord in the circle.

Chord of a circle 1 US

Let’s look at when the radius (or diameter) is perpendicular to a chord.

In circle C, \, CE is the radius and it is perpendicular to chord AB.

Chord of a circle 2 US

Notice how radius CE also bisects chord AB at a 90^{\circ} angle. Bisect means to cut in half which means that the lengths AD=BD=x

Chord of a circle 3 US

[FREE] Common Core Practice Tests (Grades 3 to 8)

[FREE] Common Core Practice Tests (Grades 3 to 8)

[FREE] Common Core Practice Tests (Grades 3 to 8)

Prepare for math tests in your state with these Grade 3 to Grade 8 practice assessments for Common Core and state equivalents. 40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!

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[FREE] Common Core Practice Tests (Grades 3 to 8)

[FREE] Common Core Practice Tests (Grades 3 to 8)

[FREE] Common Core Practice Tests (Grades 3 to 8)

Prepare for math tests in your state with these Grade 3 to Grade 8 practice assessments for Common Core and state equivalents. 40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!

DOWNLOAD FREE

Now, let’s look at the proof of this to understand why it is always true.

Using congruent triangles, you can prove this is always true.

First, draw two radii, AC and BC, to make two right triangles, triangle ADC and triangle BDC.

Chord of a circle 4 US

Since radii within a circle are congruent in length, you know that the hypotenuses of both right triangles (Triangle ADC and Triangle BDC ) are congruent.

Chord of a circle 5 US

Since both triangles share side CD, through the reflexive property, CD is congruent to itself.

Chord of a circle 6 US

At this point, you can conclude that right triangle ADC is congruent to the right triangle BDC through the right triangle congruence postulate, HL (Hypotenuse-Leg).

Since the two triangles are congruent, all of the corresponding parts of the triangle are congruent. Therefore, AD is congruent to BD, which will always be true.

Note: The radius is also the perpendicular bisector of the chord AB and the angle bisector of central angle BCA.

What is the chord of a circle?

What is the chord of a circle?

Common Core State Standards

How does this apply to geometry?

  • High School Geometry – Circles (HSG-C.A.2)
    Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

How to find missing lengths using chords

In order to find missing angles or the length of a chord

  1. Locate the key parts of the circle for an appropriate circle theorem.
  2. Use other angle or segment facts to write an equation.
  3. Solve for the missing side or angle.

Chord of a circle examples

Example 1:

In circle D, if AB=15 units, find the length of AC.

Chord of a circle 7 US

  1. Locate the key parts of the circle for an appropriate circle theorem.

  • D is the center of the circle.
  • CE is the diameter of the circle.
  • CD is the radius of the circle.
  • AB is the chord of the circle.
  • CE is perpendicular to AB.

2Use other angle or segment facts to write an equation.

Using the theorem, you know that AC=BC. Using the segment addition postulate, you know that AC+BC=AB.

3Solve for the missing side or angle.

Given that AB=15, use substitution.

\begin{aligned} AC+BC&=15 \\\\ AC&=BC \end{aligned}

\begin{aligned}BC+BC&=15 \\\\ 2BC&=15 \\\\ BC&=7.5 \end{aligned}

Example 2:

In circle F, \, NP=24 units, and EF=FM. Find the length of DE.

Chord of a circle 8 US

Locate the key parts of the circle for an appropriate circle theorem.

Use other angle or segment facts to write an equation.

Solve for the missing side or angle.

Example 3:

In circle J, \, LM=121 units and KM=3x. Find the value of x.

Chord of a circle 9 US

Locate the key parts of the circle for an appropriate circle theorem.

Use other angle or segment facts to write an equation.

Solve for the missing side or angle.

Example 4:

In circle L, \, RT=18 units and PQ=12 units. Find the length of segment LS to the nearest tenth.

Chord of a circle 10 US

Locate the key parts of the circle for an appropriate circle theorem.

Use other angle or segment facts to write an equation.

Solve for the missing side or angle.

Example 5:

In circle F, the radius length is 14 units and chord AC is 22 units, find the value of BD to the nearest tenth.

Chord of a circle 12 US

Locate the key parts of the circle for an appropriate circle theorem.

Use other angle or segment facts to write an equation.

Solve for the missing side or angle.

Example 6:

In circle O, \, AB=9 units. Find the length of BD to the nearest tenth.

Chord of a circle 14 US

Locate the key parts of the circle for an appropriate circle theorem.

Use other angle or segment facts to write an equation.

Solve for the missing side or angle.

Teaching tips for chord of circle

  • Provide opportunity for students to discover the theorems and make conjectures using compasses and protractors.

  • Use digital platforms so students can investigate strategies to problem solve using the circle theorems.

Easy mistakes to make

  • Thinking that any chord that is perpendicular to another chord bisects it
    When the diameter is perpendicular to a chord it will always bisect that chord.
    But not all chords that are perpendicular bisect each other.

    Chord of a circle 16 US
    In circle C, \, DE is the diameter which is perpendicular to chord FG. \, FG is also being bisected by DE. In circle B, chords HK and IJ are perpendicular, neither one is being bisected.

  • Not using triangles to find missing lengths
    For example, in the figure below depending on the given information, it might be helpful to sketch an isosceles triangle because segments connecting CG and CH are radii of the circle and equal in length.

    Chord of a circle 17 US

  • Using Pythagorean theorem (Pythagoras’ theorem) incorrectly
    The missing side is calculated by incorrectly adding the square of the hypotenuse and a shorter side, or subtracting the square of the shorter sides.

    For example, in the figure below finding x by writing the equation, 8^2+5^2=x^2 instead of x^2+5^2=8^2.

    Chord of a circle 18 US

  • Using the incorrect trigonometric function
    The incorrect trigonometric function is used and so the side or angle being calculated is incorrect. This also includes the inverse trigonometric functions.

    For example, using the cosine function to find segment BC instead of the sine function.

    Chord of a circle 19 US

Practice chord of a circle questions

1. In circle F, radius FG is perpendicular to chord AC. The length of the chord AC is 36 units and line segment AB is 3x. Find the value of x.

 

Chord of a circle 20 US

8
GCSE Quiz False

6
GCSE Quiz True

12
GCSE Quiz False

4
GCSE Quiz False

Since the radius is perpendicular to the chord it bisects the chord, meaning that AB=BC (so they both equal 3x ).

 

To find the value of x, you can apply the segment addition postulate,

 

\begin{aligned}3x+3x&=36 \\\\ 6x&=36 \\\\ x&=6 \end{aligned}

2. In circle G, \, GI is perpendicular to AD. If AB=3x-1 and BD=2x+7, find the length of AD.

 

Chord of a circle 21 US

23 units

GCSE Quiz False

8 units

GCSE Quiz False

24 units

GCSE Quiz False

46 units

GCSE Quiz True

GI is perpendicular to AD which means it also bisects AD, meaning that AB=BD.

 

The equation to solve for x is:

 

3x-1=2x+7

 

Subtract 2x from both sides of the equation

 

x-1=7

 

Add 1 to both sides of the equation to isolate x.

 

x=8

 

Substitute the value of x into the expression to find the value of AD.

 

AD=BD=3(8)-1=24-1=23

 

2(AB)=AD so AD=2\times{23}=46 units

3. In circle C, the radius is 10 units and AD is 12 units. Find the length of CF.

 

Chord of a circle 22 US

6 units

GCSE Quiz False

8 units

GCSE Quiz True

10 units

GCSE Quiz False

4 units

GCSE Quiz False

Draw in another radius to create the right triangle AFC.

 

Chord of a circle 23 US

 

Then using the Pythagorean Theorem, solve for CF.

 

The radius is the hypotenuse of the triangle which is 10 units.

 

AF is half the measure of AD so it is 6 units (\cfrac{1}{2}\times{12}=6)

 

\begin{aligned}(CF)^2+6^2&=10^2 \\\\ (CF)^2+36&=100 \\\\ (CF)^2&=64 \\\\ CF&=8 \end{aligned}

 

CF=8 units

4. In circle G, \, RT is perpendicular to AB. If RT=22 and AB=16, find HT. Round the answer to the nearest tenth.

 

Chord of a circle 24 US

3.5 units

GCSE Quiz True

7.5 units

GCSE Quiz False

8 units

GCSE Quiz False

11 units

GCSE Quiz False

The diameter, RT, is perpendicular to chord AB which means that it also bisects AB making AH=BH.

 

If AB=16, then AH=BH=\cfrac{1}{2}\times{16}=8

 

The diameter length is 22 so the radius length AG=\cfrac{1}{2}\times{22}=11.

 

AG=11

 

Draw in a radius to make the right triangle GAH.

 

Chord of a circle 25 US

 

Using the Pythagorean theorem, you can write the equation,

 

\begin{aligned}(GH)^2+8^2&=11^2 \\\\ (GH)^2+64&=121 \\\\ (GH)^2&=57 \\\\ GH&\approx{7.5} \end{aligned}

 

GT is a radius of circle G.

 

The radius length is 11, from the center of the circle at point G to H is 7.5.

 

To find HT subtract the two lengths.

 

\begin{aligned}HT&=GT-GH \\\\ HT&=11-7.5 \\\\ HT&=3.5 \end{aligned}

 

HT=3.5 units

5. In circle O, find the length of AE, if BD is perpendicular to AC and chord CD=8\sqrt{3}.

 

Chord of a circle 26 US

2\sqrt{3} units

GCSE Quiz False

8 units

GCSE Quiz False

4\sqrt{3} units

GCSE Quiz True

4 units

GCSE Quiz False

The diameter, BD is perpendicular to chord AC which means it also bisects AC, so AE=CE.

 

Using the right triangle CED, you can solve for x which is CE.

 

The right triangle is a 30-60-90 triangle with the hypotenuse, CD, being equal to 8\sqrt{3}.

 

Using the ratios of the sides of a 30-60-90, you can solve for x.

 

\cfrac{\text{short side}}{\text{hypotenuse}}=\cfrac{1}{2}=\cfrac{x}{8\sqrt{3}}

 

\begin{aligned}\cfrac{1}{2}&=\cfrac{x}{8\sqrt{3}} \\\\ 2x&=8\sqrt{3} \\\\ x&=4\sqrt{3} \end{aligned}

 

Segment CE=x, so CE=4\sqrt{3}

 

Since AE=CE, then AE=4\sqrt{3}

6. AC is the diameter of circle O and is perpendicular to BD. If segment DE is 7\mathrm{~cm} and angle BAE=22^{\circ}, find the value of x to the nearest tenth.

 

Chord of a circle 27 US

18.7\mathrm{~cm}
GCSE Quiz True

2.6\mathrm{~cm}
GCSE Quiz False

7.5\mathrm{~cm}
GCSE Quiz False

6.5\mathrm{~cm}
GCSE Quiz False

Since AC is the diameter of a circle and perpendicular to chord BD, it also bisects BD, so BE=DE.

 

You are given that DE=7\mathrm{~cm} which means that BE=7\mathrm{~cm}.

 

Using the right triangle BEA, use the sine ratio to find the value of x.

 

Chord of a circle 28 US

 

\begin{aligned}\sin(22)&=\cfrac{7}{x} \\\\ x\sin(22)&=7 \\\\ x=\cfrac{7}{\sin(22)}&\approx{18.7} \\\\ AB&=18.7\mathrm{~cm} \end{aligned}

Chord of a circle FAQs

What is a secant line to a circle?

The secant line is a line that intersects a circle at two points. The two points where the secant intersects a circle are on the edge of the circle or the circumference of the circle. Chords are part of a secant line.

What is the arc length?

The arc length is the actual distance of length of an arc of a circle. It is a portion of the circumference of the circle. Not to be confused with the degree measurement of the arc.

Do equal chords of a circle subtend or intercept equal angles?

Equal chords of a circle subtend equal arcs (minor arcs) not angles.

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