Theorems on circles and other terms related to circles.

2. Have you ever wondered how architects and engineers use
circles?
How important are circles in the field of transportation,
industries, sports, navigation, carpentry, and in your daily
life?

3. The Provincial Disaster and Risk Reduction Management Committee (PDRRMC)
advised the residents living within the 14-km radius critical area to evacuate due to
eminent eruption of a volcano. If the map is drawn on a coordinate plane, the
coordinates corresponding to the location of the volcano is (4, 5).

4. CIRCLES
A circle is the set of all points in a plane
that are equidistant from a given point
in the plane known as the center of the
circle.

5. A radius (plural, radii) is a segment from the
center of the circle to a point on the circle.
A chord is a segment whose endpoints lie on a
circle.
A diameter is a chord that contains the center
of the circle.

6. Radius = PO
Diameter = MO
Chord = SE

7. An arc is an unbroken part of a circle.
Any two distinct points on a circle divide
the circle into two arcs. The points are
called the endpoints of the circle.

9. Arcs
 A semicircle is an arc whose endpoints are endpoints of a
diameter. A semicircle is informally called a half-circle. A
semicircle is named by its endpoints and another point that lies
on the arc.
 A minor arc of a circle is an arc that is shorter than a semicircle
of that circle.
 A major arc is named by its endpoints. A major arc of a circle is
an arc that is longer than a semicircle of that circle. A major arc
is named by its endpoints and another point that lies on the
arc.

10. Central Angle
A central angle is an angle in the plane of a
circle whose vertex is the center of the circle.
An arc whose endpoints lie on the sides of the
angle and whose other points lie in the interior
of the angle is the intercepted arc of the
central angle.

18. Learning Target:
I can derive inductively the relations among center,
radius, chord, and diameter of a circle.

19. Theorem 104
 If a radius is perpendicular to a chord, then it bisects
the chord.
Given: Circle O with radius 𝑂𝐷 , 𝑂𝐷 ⊥ 𝐴𝐵 at E.
Prove: 𝑂𝐷 bisects 𝐴𝐵.
O
A
F B
D

20. Given: Circle O with radius 𝑂𝐷 , 𝑂𝐷 ⊥ 𝐴𝐵 at E.

21. Theorem 105
 If the radius of a circle bisects a chord that is not a
diameter, then it is perpendicular to the chord.

22. Theorem 106
 The perpendicular bisector of a chord passes through
the center of the circle.
Given:𝑃𝑄 is the perpendicular bisector of 𝐴𝐵.
O
A
Q
B
P
Prove: 𝑃𝑄 passes through O.

23. O
A
Q
B
P

24. Concentric Circles are coplanar circles having the same center.
Circle O with radius 𝑂𝐴 and circle O with radius 𝑂𝐵 below are
concentric circles.

25. Theorem 107
 If chords of a circle or of congruent circles are equidistant from
the center(s), then the chords are congruent.
CHORDS EQUIDISTANT FROM THE CENTER
O
A C B
P
D F E
Given: Circle O ≅ Circle P; OC = PF; 𝑂𝐶 ⊥ 𝐴𝐵 and 𝑃𝐹 ⊥ 𝐷𝐸.
Prove: 𝐴𝐵 ≅ 𝐷𝐸

26. Theorem 108
 If chords of a circle or of congruent circles are
congruent, then they are equidistant from the centers
of the circles.
CHORDS EQUIDISTANT FROM THE CENTER

27. CHORDS EQUIDISTANT FROM THE CENTER

28. CHORDS EQUIDISTANT FROM THE CENTER

29. CHORDS EQUIDISTANT FROM THE CENTER
PQ = 4 OS
PS = 5

30. CHORDS EQUIDISTANT FROM THE CENTER
PQ = 4 OS
PS = 5

31. CHORDS EQUIDISTANT FROM THE CENTER
PQ = 4 OS
PS = 5

32. Theorem 106
 The perpendicular bisector of a chord passes through
the center of the circle.