# 6.2 – Chords

Key Terms

• Chord – Any line segment whose endpoints are on the circle.
• Diameter – A line segment that contains (passes through) the center of the circle and has endpoints on the circle.
• The diameter is a chord, as it has two endpoints ON the edge of the circle.
• This term also refers to the length of this line segment
• The diameter of a circle is twice the length of the radius.

Review

• Circumference – the distance around a circle
• To find the circumference, use $2\pi r$ or $\pi d$

• Perpendicular Bisector
• A line, ray, or segment
• Intersects a segment at its midpoint
• Forms right angles
• Divides a segment in half
• Is the shortest path between a line segment and a point not on that segment

Notes

• Chords
• Watch the video and draw an circle with chord $\overline{AB}$.

• Congruence and Equidistance
• If two chords are the same distance from the center of a circle, then they must be congruent.
• If two chords of a given circle are congruent, then they must be equidistant from the center of the circle.

• The shortest path between a line and any point (not on the line) is a path that is perpendicular to the line.
• If a radius of a circle bisects a chord, then it is perpendicular to that chord.
• If a radius of a circle is perpendicular to a chord, then it bisects that chord.

• Conditional Statement Proof
• Again, if a radius of a circle is perpendicular to a chord, then it bisects that chord.
• Prove that the Radius $\overline{OC}$ Bisects Chord $\overline{AB}$

• Converse Statement Proof (true in this case)
• Again, if a radius of a circle bisects a chord, then it is perpendicular to that chord.
• Prove that the Radius $\overline{OC}$ is Perpendicular to Chord $\overline{AB}$

• Template for Proofs

• Diameter
• Another name for a chord that passes through the center of a circle is the diameter.
• Changing the diameter of a circle changes the size of the circle because the diameter is two radii; and, the radius defines the size of the circle.

Examples

• Ex 1. In $\odot{O}$, radius $\overline{OP}$ intersects chord $\overline{AC}$ in point B so that AB = 8 units and BC = 8 units.
• This means that $\overline{OP}$ is perpendicular to $\overline{AC}$.
• Try drawing this and marking congruency.

• Ex 2. Given $\odot{V}$ below, you can conclude that $\overline{QS}$ is congruent to $\overline{NR}$

• Ex 3. Given $\odot{M}$ below, you can conclude that $\overline{ML}$ is congruent to $\overline{MA}$

• Ex 4. If the blue radius below is perpendicular to the green chord and the segment $\overline{AB}$ is 9 units long, what is the length of the chord?
• Since $\overline{AC}$ has been bisected by the radius of the circle, $\overline{AB}$ is half of $\overline{AC}$.
• $\overline{AB}$ and $\overline{BC}$ are congruent, so $\overline{BC}$ also equals 9.
• 9 + 9 = 18, so $\overline{AC}=18$.

• Ex 5. The blue segment below is a radius of $\odot{O}$. What is the length of the diameter of the circle?
• Since the diameter is twice the length of the radius: 9.3(2) = 18.6 units is the length of the diameter.

• Ex 6. The blue segment below is a diameter of $\odot{O}$. What is the length of the radius of the circle?
• Since the diameter is twice the length of the radius: $\frac{7.7}{2}=3.85$ units is the length of the radius.