Line (Segment) from the Center to a Chord

Line(Segment) from the Center to a Chord

Let us investigate a line (line segment) drawn from the center of the circle to a chord. BC is a chord of the circle with center at O. D is a point on the chord. What do you notice about the lengths BD and CD when the angle is 90℃.
Move the point D on the chord and look for a situation when the angle is 90℃ at this point what can you say about the lengths BD and CD, are they equal? You will observe that BD = CD. Here we can make the following conjecture:

• The perpendicular from the center of a circle to a chord bisects the chord.

The converse of the above conjecture is also true:

• If a line (line segment) is drawn from the center of a circle to the midpoint of a chord, then the line is perpendicular to the chord.

Let us prove the conjecture 1 above. This proof is based on the congruency of the triangles.

       Given: OD ⊥ BC

       Construction: Join OB and OC.

       Strategy: If we can show that △BDO 
       and △CDO are congruent then the 
       sides BD and CD must be equal.

       To prove: BD = CD

       Proof:

          In △ BDO and △ CDO

          OB = OC (Radii of the circle)

          OD = OD (Common Sides)

         ∠BDO = ∠CDO (given , both 90)

      Hence, △ BDO ≅ △ CDO

      Therefor, BD = CD (Corresponding Parts of Congruent Triangles)

Angles - Line in Space

Perpendicular to a Line - 4

Following are the steps for construction of a perpendicular to a line from a point outside it. In this construction, the arcs are shown as circles. 

1. Let AB be the given line and P is the point outside it 
2. With A as centre and radius AP, draw an arc cutting the given line at C. 
3. With C as centre and radius CP, draw an arc cutting the arc of step 2 at R. 
4. Draw line segment joining PR intersecting AB at Q.

PQ is the required perpendicular. Drag Point P and see the effect. This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Reflection About a Line

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com