We need to prove that OX YC (why?), this must mean that OX and OY are perpendiculars to the two chords from the center (this means that they will respectively bisect the two chords as well). The following figure shows two chords AB and CD of a circle with center O, such that AB > CD. As your chord moves closer and closer to the center, it increases in length, as the following figure shows (the diameter is the largest possible chord in any circle): Visually speaking, this should be obvious. Theorem: For two unequal chords of a circle, the greater chord will be nearer to the center than the smaller chord. We now talk about a more general result related to unequal chords. Thus, the two chords are of equal length. Thus, since it is given that AB = CD, we have Similarly, since OY is perpendicular to CD, it must also bisect CD. The segments formed on a chord when two chords of a circle intersect 3. If two chords of the same circle or congruent circles are congruent, then their corresponding arcs are congruent. Note that since OX is perpendicular to AB, it must also bisect AB. Chords Vocabulary Match each definition with its corresponding term. The distances of the two chords are respectively OX and OY: The following figure shows two equidistant chords AB and CD of a circle with center O. Theorem: Equal chords of a circle are equidistant from its center. "Chords equidistant from the center of circle are equal in Algebraden.Now, we discuss some other fundamental results related to circles. Hence, this proves property 6 of circle i.e. Put the values from above statement 2 & 3 and we get: Since, we know that corresponding parts of congruent triangles are equal, so we get:
Therefore, on applying RHS Rules of congruency, we get: OQ = OR (radii of circle are always equal) Now, join points O & Q and O & R (as shown below):Īngle 1 = Angle 2 (90 degree each - proved in above statement 1) Similarly, O is the center of circle (given)Īnd OB is perpendicular to RS (proved in above statement 1) Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres). So apply Property 3 of circle, "The perpendicular from the center of a circle to a chord bisects the chord" and we get: (statement 1)Īnd OA is perpendicular to PQ (proved in above statement 1) equidistant chords theorem in the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. OA is perpendicular to PQ and OB is perpendicular to RS (as shown below). If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter. Since OA and OB is the distance of chords PQ & RS respectively from the center of circle.Īnd as per the property which says "The length of perpendicular from a point to a line is the distance of line from the point", so we get: Property : Perpendicular from the center of a circle to a chord bisects the chord What are Corresponding Parts of Congruent Triangles ? What is RHS Rule of congruency in Triangles ? Study with Quizlet and memorize flashcards containing terms like 1) A is the locus of points in a plane that are all equidistant from a single point., 2) is a mathematical constant that is equal to the ratio of the circumference of a circle to its diameter., 3) The point in the interior of the circle that is equidistant from every point on the circle is called the of the.
EQUIDISTANT CHORD GEOMETRY HOW TO
How to prove this property : Chords equidistant from the center of circle are equal in lengthīefore you prove this property, you are advised to read: Now, as per the property 6 of circle i.e."Chords equidistant from the center of circle are equal in length", we get: OA and OB is the distance of chords PQ & RS respectively from the center of circle What is Distance of a Line from the point ? Line drawn from the center of circle to bisect a chord, is perpendicular to the chordĮqual chords are equidistant from the center of circleĬhords equidistant from the center of circle are equal in lengthīefore you understand the property "Chords equidistant from the center of circle are equal in length", you are advised to read: Perpendicular from the center of a circle to a chord bisects the chord NOTE: Two chords are said to be congruent if they are equal in length. If Angles subtended by the chords at the center of circle are equal, then chords are also equal Two congruent chords of a circle are equidistant from its center. Home > Circle > Properties of Circle > Chords equidistant from the center of circle are equal in length > Chords equidistant from the center of circle are equal in lengthĮqual chords subtend equal angles at the center of a circle
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