Lesson The longer is the chord the larger its central angle is

Algebra ->  Circles -> Lesson The longer is the chord the larger its central angle is      Log On


   

This Lesson (The longer is the chord the larger its central angle is) was created by by ikleyn(52765) About Me : View Source, Show
About ikleyn:

The longer is the chord the larger its central angle is



In this lesson you will find the proofs to the following statements:

    1)  the longer is a chord the larger its central angle is;
    2)  the larger is a central angle the longer the chord is;
    3)  if two chords in a circle are congruent then their corresponding central angles are congruent;
    4)  if two central angles in a circle are congruent then their corresponding chords are congruent.

Theorem 1

The longer is a chord in a circle the larger the corresponding central angle is.

(Surely,  when we are talking on the corresponding central angle to a given chord we mean the lesser of the two central angles).

Proof

We are given a circle with the center  P  and two chords  AB  and  CD                        
in the circle  (see the  Figure 1a  or the  Figure 1b),  where  CD 
is longer than  AB.  Let us draw the radii connecting the endpoints of
the chords with the center of the circle.  We need to prove that the
central angle  LCPD  is greater than the central angle  LAPB.

For the proof,  let me remind you the  SSS inequality theorem:

    If two sides of one triangle are congruent to the corresponding two
    sides of another triangle,  but the third side of the first triangle
    is longer than the third side of the second triangle,  then the angle
    included between the two sides of the first triangle is greater than
    the angle included between the two sides of the second triangle.



Figure 1a.  To the Theorem 1      



Figure 1b.  To the Theorem 1

It was proved in the lesson  Angles and sides inequality theorems for triangles  under the topic  Triangles  of the section  Geometry  in this site.

Now consider the triangles  DELTACPD  and  DELTAAPB.  They both are isosceles triangles and their lateral sides are congruent to the radius of the circle.
Therefore,  the  Theorem 1  directly follows the  SSS inequality theorem.
The proof is completed.

Theorem 2

The larger is a central angle in a circle the longer the corresponding chord is.

(We consider here central angles that are lesser the straight angle (180°)).

Proof

We are given a circle with the center  P  and two central angles  LAPB                          
and  LCPD  in the circle  (see the  Figure 2a  or the  Figure 2b),  where
LCPD  is greater than  LAPB.  We need to prove that the chord  CD  is
longer than the chord  AB.

For the proof,  use the  SAS inequality theorem:

    If two sides of one triangle are congruent to the corresponding two sides
    of another triangle,  and the included angle of the first triangle is greater
    than the included angle of the second triangle,  then the third side of the
    first triangle is longer than the third side of the second triangle.



Figure 2a.  To the Theorem 2      



Figure 2b.  To the Theorem 2

It was proved in the lesson  Angles and sides inequality theorems for triangles  under the topic  Triangles  of the section  Geometry  in this site.

Now consider the triangles  DELTACPD  and  DELTAAPB.  They both are isosceles triangles and their lateral sides are congruent to the radius of the circle.
Therefore,  the  Theorem 2  directly follows the  SAS inequality theorem.
The proof is completed.

Theorem 3

If two chords in a circle are congruent then their corresponding central angles are congruent.

Proof

The proof is similar to that of the  Theorem 1.
Instead of referring to the  SSS inequality theorem,  simply use the  SSS test for the triangles congruency.

Theorem 4

If two central angles in a circle are congruent then their corresponding chords are congruent.


Proof

The proof is similar to that of the  Theorem 2.
Instead of referring to the  SAS inequality theorem,  simply use the  SAS test for the triangles congruency.

Summary

Two chords in a circle are congruent if and only if their corresponding central angles are congruent.
Two chords in a circle are congruent if and only if their corresponding arcs are congruent.
In a circle,  the longer chord has larger corresponding central angle and longer corresponding arc;  the shorter chord has smaller corresponding central angle and shorter corresponding arc.


My other lessons on circles in this site are
    - A circle, its chords, tangent and secant lines - the major definitions,
    - The chords of a circle and the radii perpendicular to the chords,
    - A tangent line to a circle is perpendicular to the radius drawn to the tangent point,
    - An inscribed angle in a circle,
    - Two parallel secants to a circle cut off congruent arcs,
    - The angle between two chords intersecting inside a circle,
    - The angle between two secants intersecting outside a circle,
    - The angle between a chord and a tangent line to a circle,
    - Tangent segments to a circle from a point outside the circle,
    - The converse theorem on inscribed angles,
    - The parts of chords that intersect inside a circle,
    - Metric relations for secants intersecting outside a circle  and
    - Metric relations for a tangent and a secant lines released from a point outside a circle
under the current topic,  and
    - HOW TO bisect an arc in a circle using a compass and a ruler,
    - HOW TO find the center of a circle given by two chords,
    - Solved problems on a radius and a tangent line to a circle,
    - Solved problems on inscribed angles,
    - A property of the angles of a quadrilateral inscribed in a circle,
    - An isosceles trapezoid can be inscribed in a circle,
    - HOW TO construct a tangent line to a circle at a given point on the circle,
    - HOW TO construct a tangent line to a circle through a given point outside the circle,
    - HOW TO construct a common exterior tangent line to two circles,
    - HOW TO construct a common interior tangent line to two circles,
    - Solved problems on chords that intersect within a circle,
    - Solved problems on secants that intersect outside a circle,
    - Solved problems on a tangent and a secant lines released from a point outside a circle
    - The radius of a circle inscribed into a right angled triangle
    - Solved problems on tangent lines released from a point outside a circle
under the topic  Geometry  of the section  Word problems.

The overview of lessons on Properties of Circles is in this file  PROPERTIES OF CIRCLES, THEIR CHORDS, SECANTS AND TANGENTS.
You can use the overview file or the list of links above to navigate over these lessons.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link  GEOMETRY - YOUR ONLINE TEXTBOOK.

This lesson has been accessed 6581 times.