G10 Math Q2- Week 6- Proves theorems on secant and tangent.pptx

3. Theorems on Secant
and Tangent

4. Lesson targets:
Determine the tangent segments and
secant segments of a circle
Solve problems involving tangents and
secants of circles.
Realize the importance of the topic in
real-life situations.

5. 18
48
15
Warm Up
Solve for x.
1. X = 9
4 2
2. 3x = 122
3. 3(x) = (15)(3)

6. 18
48
14
Warm Up
Solve for x.
4. 2X = 9
4 2
5. x2 = 49
6. BC and DC are tangent
to A. Find BC.

7. Theorem 1.
If two chords of a circle intersect,
then the product of the measures
of the segments of one chord is
equal to the product of the
measures of the segments of the
other chord.

9. Example 1: Applying the Chord-Chord
Product Theorem
Find the value of x.
EP  PK = SP  AP
x(8) = 4(6)
8x = 24
x = 3
E
S
A
K
6
4
x
P
8

10. Example 2: Applying the Chord-Chord
Product Theorem
Find the value of x and the length of
each chord.
EJ  JF = GJ  JH
10(7) = 14(x)
70 = 14x
5 = x
EF = 10 + 7 = 17
GH = 14 + 5 = 19
J J

11. Theorem 2.
If two secant segments are drawn to a circle
from an exterior point, then the product of
the lengths of one secant segment and its
external secant segment is equal to the
product of the lengths of the other secant
segment and its external secant segment. .

12. A secant segment is a segment of a secant
with at least one endpoint on the circle.
An external secant segment is a secant
segment that lies in the exterior of the circle
with one endpoint on the circle.

14. Find the value of x and the length of each secant
segment.
Example 1: Applying the Secant-Secant
Product Theorem
16(7) = (8 + x)8
112 = 64 + 8x
48 = 8x
6 = x
ED = 7 + 9 = 16
EG = 8 + 6 = 14

15. Find the value of z and the length
of each secant segment.
39(9) = (13 + z)13
351 = 169 + 13z
182 = 13z
14 = z
LG = 30 + 9 = 39
JG = 14 + 13 = 27
Example 2: Applying the Secant-Secant
Product Theorem

16. Theorem 3.
If two segments from the same
exterior point are tangent to a
circle, then the two segments
are congruent.

17. A tangent segment is a segment of a
tangent with one endpoint on the
circle. AB and AC are tangent
segments.
If the two tangent
segments are
drawn to a circle
from external
point, then the two
tangent segments
are congruent.

18. BC and DC are tangent
to A. Find BC.
BC = DC
5y – 1 = 3y + 5
5y – 3y = 5 + 1
2y = 6
2 2
y = 3
BC = 5y – 1
BC = 5(3) – 1
BC = 15 – 1
BC = 14

19. Theorem 4.
If a tangent segment and a secant segment
are drawn to a circle from an exterior point,
then the square of the length of the
tangent segment is equal to the product of
the lengths of the secant segment and its
external secant segment.

21. Example 4: Applying the Secant-Tangent
Product Theorem
Find the value of x.
The value of x must be 10 since it represents a
length.
ML  JL = KL2
20(5) = x2
100 = x2
±10 = x

22. Find the value of y.
DE  DF = DG2
7(7 + y) = 102
49 + 7y = 100
7y = 51

23. Activity 3
Shaped Me!

24. FIND ME!
1. Find the value of d and the
length of each chord.
3. Find the value of a.
2. Find the value of x and the
length of each secant segment.
4. Find the value of y.
S H I
T
F
y
9
8
10

26. Time is
up!

27. FIND ME!
1. Find the value of d and the
length of each chord.
3. Find the value of a.
2. Find the value of x and the
length of each secant segment.
4. Find the value of y.
S H I
T
F
y
9
8
10
d = 9
ZV = 17
WY = 18
x = 10
QP = 8
QR = 12
a = 8
y = 16

28. How are you going to
solve problems involving
tangents and secants of
circles?

31. If the two tangent segments are
drawn to a circle from external
point, then the two tangent
segments are congruent.

33. Individual Written Activity
1. Find the value of GE in the figure at the right?
A. 8 units C. 22 units
B. 72 units D. 4x units
3. Apply theorem number 4 in finding the value
of x in the figure at the right.
A. 25 units C. 15 units
B. 16 units D. 225 units
2. In the figure at the right, SI and TI are two
secants, find the value of the length of HI.
A. 144 units C. 16 units
B. 9 units D. 18 units

34. Time is
up!

35. ASSIGNMENT. Understand Me More…
Answer the following questions. Use the rubric provided
to rate your work.
Jurene and Janel were asked to find the length of AB in
the figure below. The following are their solutions.

36. Thank you for you
participation!
See you around!

37. Individual Written Activity
1. Find the value of x and the
length of each chord.
x = 18 GI = 22 DU = 16
x = 𝟗 SI = 16 TI = 18
3. Find the value of x.
15
2. Find the value of x and the length
of each secant segment.