grade 10 Math lesson

1. Relationship Between Chords and
Arcs
A. Bisected Chord Theorem
B. Apothem
C. Congruent Chords Theorem

2. A. Bisected Chord Theorem
If a radius of a circle bisects a chord on the same circle, then the radius and the
chord are perpendicular.
Example:
In the figure to the right,
the radius 𝐴𝐵 bisects
the chord 𝐶𝐷 at 𝑋. It
follows that
𝐴𝐵 ⊥ 𝐶𝐷.

3. A. Bisected Chord Theorem
1. point of bisection or
midpoint X of the
bisecting radius 𝑨𝑩
2. As a result,
a. segment 𝐶𝑋 = 𝑋𝐷
b. arc 𝑪𝑩 = 𝑩𝑫 and
c. radius 𝑨𝑩 ⊥ 𝑪𝑫

4. A. Bisected Chord Theorem
1. point of bisection or
midpoint X of the
bisecting radius 𝑨𝑩
2. segment 𝐶𝑋 and 𝑋𝐷
3. arc 𝑪𝑩 and 𝑩𝑫

5. Learn about It!
2 Apothem
a distance of a chord to the center of the circle
Example:
In the figure to the right, 𝑉𝑇
represents the distance between
𝑈𝑊 and the center 𝑇. As such 𝑉𝑇
is the apothem. In the same
manner, 𝑇𝑌 is also the apothem
between 𝑋𝑍 and the center 𝑇.

6. B. Apothem
apothem
𝑽𝑻 and 𝑻𝒀

7. Learn about It!
3 Congruent Chords Theorem
If two chords of a circle are congruent, then their apothems and
intercepted arcs are also congruent.
Example:
The figure to the right illustrates
circle 𝑂 with congruent chords 𝐴𝐵
and 𝐷𝐸. It follows that the
apothems 𝐹𝑂 and 𝐶𝑂 are
congruent as well as the
intercepted arcs 𝐴𝐵 and 𝐷𝐸.

8. C. Congruent Chords Theorem
1. chord
𝐴𝐵 and 𝐷𝐸
2. arc 𝐴𝐵 and 𝐷𝐸
if 𝐴𝐵 = 𝐷𝐸
also therefore,
𝐴𝐵 = 𝐷𝐸

9. Example 1: In a given
circle 𝑂, the chords
𝐴𝐵 and 𝐵𝐶 are
congruent. What can
be concluded
between 𝐴𝐵 and 𝐵𝐶?

10. Try It!
Example 1: In a given circle 𝑂, the chords 𝐴𝐵 and
𝐵𝐶 are congruent. What can be concluded between
𝐴𝐵 and 𝐵𝐶?
Solution: Let us illustrate
the problem.
Since 𝐴𝐵 ≅ 𝐵𝐶, it follows
that 𝑨𝑩 ≅ 𝑩𝑪 according to
the Congruent Chords
Theorem.

11. Example 2: In circle 𝑂, the chords 𝐴𝐵 and 𝐶𝐷 are
congruent. If 𝑚 𝐴𝐵 = 2𝑥 + 13 ° and 𝑚 𝐶𝐷 = 3𝑥

12. Example 2: In circle 𝑂, the chords 𝐴𝐵 and 𝐶𝐷 are
congruent. If 𝑚 𝐴𝐵 = 2𝑥 + 13 ° and 𝑚 𝐶𝐷 = 3𝑥 − 2 ° ,
find 𝑚 𝐴𝐵 and 𝑚 𝐶𝐷.
Solution 2.1: Since 𝑚 𝐴𝐵 = 𝑚 𝐶𝐷, we can solve for
the value of 𝑥 by substituting their corresponding
values.
Thus,
𝑚 𝐴𝐵 = 𝑚 𝐶𝐷
2𝑥 + 13 = 3𝑥 − 2
2𝑥 − 3𝑥 = −2 − 13
−𝑥 = −15
𝑥 = 𝟏𝟓

13. Example 2: ….. If 𝑚 𝐴𝐵 = 2𝑥 + 13 ° and 𝑚 𝐶𝐷
= 3𝑥 − 2 ° ,find 𝑚 𝐴𝐵 and 𝑚 𝐶𝐷.
Solution 2:
𝑚 𝐴𝐵 = 2𝑥 + 13
= 2 15 + 13
= 30 + 13
= 𝟒𝟑
= 43°

14. ACTIVITY 1
1. a radius bisected bisected a 4 inches
chord 𝐴𝐵 at midpoint Q, what would
be measure of 𝐴𝑄 and 𝑄𝐵?
2. In circle 𝑂, the radius 𝑂𝐴 passes through the
chord 𝐵𝐶 at its midpoint 𝐷. What relationship
can be said between 𝑂𝐴 and 𝐵𝐶?
3. if the chord 𝑆𝐵 whose apothem is 2 inches and
whose arc is 75⁰, is said to congruent with 𝐶𝐸, what
would the measure of the apothem and arc of 𝐶𝐸?

15. ACTIVITY 2

16. ACTIVITY 3. Solve the following word problems.

17. ACTIVITY 3. Solve the following word problems.