10.2 Arcs and Chords Unit IIIC Day 3

Do Now How do we measure distance from a point to a line? The distance from a point to a line is the length of the perpendicular segment from the point to the line.

Theorem 10.4 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. ◦  if and only if

Ex. 4: Using Theorem 10.4 Find m.

Using Chords of Circles A point Y is called the midpoint of if . Any line, segment, or ray that contains that point Y bisects.

Theorem 10.5 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. ◦ ___ ≅ ___, ___ ≅ ___

Theorem 10.6 (converse of 10.5) If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. ◦ ______ is a diameter.

Ex. 5: Finding the Center of a Circle Step 1: Draw any two chords that are not parallel to each other. Step 2: Draw the perpendicular bisector of each chord. These must be __________. Step 3: The diameters must intersect at the circle’s ________.

Ex. 6: Using Properties of Chords The top of a window is in the shape of a circular arc, as shown. Find the center of the circle used to form the arc.

Ex. 6A: Using Properties of Chords The pick side of a masonry hammer works best if you swing it along a circular curve that matches the shape of the pick. Find the center of the circular swing.

Theorem 10.7 In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. AB  CD if and only if _____________

Ex. 7: Using Theorem 10.7 AB = 8; DE = 8, and CD = 5. Find CF.

Closure Find the length of a chord of a circle with radius 8 that is a distance of 5 from the center.