MIDLINE THEOREM LESSON PLAN

3. Classroom Rules
Listen when someone is talking.
Raise your hand to speak, or to
get up out of the chair.
Follow directions when first
giving.
Be respectful to each other
and work quietly.

4. QUAD is a square. Use the given figure to answer the following
and state its property that justify your answer.
1. QA = 17, QE =
2. DA = 37, UA =
3. 𝑚∠𝑄𝑈𝐴 =
4. 𝑚∠𝑄𝐸𝐷 =
5. 𝑚∠𝑈𝐷𝐴 =
6. If QE = 13. What is
the sum of the two diagonals?
A
U
Q D
E
QE = 8.5, Diagonals
bisect each other
UA = 37, All sides are
congruent
𝒎∠𝑸𝑼𝑨 = 𝟗𝟎𝟎
, It has four
right angles
𝒎∠𝑸𝑬𝑫 = 𝟗𝟎𝟎
, Diagonals of a
square are perpendicular
𝒎∠𝑼𝑫𝑨 = 𝟒𝟓𝟎
, Each diagonal of
a square bisects opposites
angles.
Sum of two diagonals is
52, diagonals of a square
are congruent.

5. Recall the following terms by choosing its respective
example inside the box provided below.
1. Midpoint
2. Transitive Property
3. Reflexive Property
4. Segment Addition Postulate
𝐴𝐷 ≅ 𝐴𝐷
𝐴𝐵 ≅ 𝐶𝐷 and 𝐴𝐵 ≅ 𝑋𝑌 then
𝐶𝐷 ≅ 𝑋𝑌
𝑈𝑇 + 𝑇𝐸 = 𝑈𝐸
If T is the midpoint of 𝑀𝐼 then 𝑀𝑇 = 𝑇𝐼
If T is the midpoint of 𝑀𝐼 then
𝑀𝑇 = 𝑇𝐼
𝐴𝐵 ≅ 𝐶𝐷 and 𝐴𝐵 ≅ 𝑋𝑌
then 𝐶𝐷 ≅ 𝑋𝑌
𝐴𝐷 ≅ 𝐴𝐷
𝑈𝑇 + 𝑇𝐸 = 𝑈𝐸

6. MAGIC TRIANGLE
To solve a magic triangle, arrange the numbers (1-6) for each triangle
so that the sum of numbers on each side is equal to the sum of
numbers on every other side.
1
6
2
3
5
4 4
2
6
5
3
1
9 11

8. At the end of the lesson the
students should be able to:
a. state the midline theorem.
b. prove/verify and apply the
midline theorem.
c. appreciates the relevance of the
topic to real-life situation.
LEARNING OBJECTIVES

9. 1. The class will be divided into 4 groups
with 10 – 12 members each.
2. Half of the groupings will do the proving and
the other half will do the verifying of the midline
theorem on triangle.
3. 10 minutes will be allotted for each group to
perform the assigned task.
4. Remember the word GROUPS.
Setting of Standards

10. GROUP ACTIVITY

11. For proving:
Directions: Given a two-column proof, supply the missing
part/statement to prove the midline theorem.
Given: ∆𝐻𝑁𝑆, 𝑂 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐻𝑁, E is the midpoint
of 𝑁𝑆
Prove: 𝑂𝐸 ∥ 𝐻𝑆, 𝑂𝐸 =
1
2
𝐻𝑆
N
H
2
O
S
3
E
4
T
1

12. STATEMENTS REASONS
1. ∆𝐻𝑁𝑆, O is the midpoint of 𝐻𝑁,
E is the midpoint of 𝑁𝑆
1.
2. In a ray opposite 𝐸𝑂, there
is a point T such that OE = ET
2. In a ray, point at a given
distance from the endpoint of
the ray.
3. 𝐸𝑁 ≅ 𝐸𝑆 3.
4. ∠2 ≅ ∠3 4.
5. ∆𝑂𝑁𝐸 ≅ ∆𝑇𝑆𝐸 5.
6. ∠1 ≅ ∠4 6.
7. 𝐻𝑁 ∥ 𝑆𝑇
7.
8. 𝑂𝐻 ≅ 𝑂𝑁 8.
9. 𝑂𝑁 ≅ 𝑇𝑆 9.
Given
Definition of midpoint
Vertical Angles Theorem
SAS Congruence Postulate
CPCTC
If alternate interior angles are
congruent, then the lines are
parallel
Definition of Midpoint
CPCTC (SN 5)

13. STATEMENTS REASONS
10. 𝑂𝐻 ≅ 𝑆𝑇 10.
11. Quadrilateral HOTS is a
parallelogram.
11.
12. 𝑂𝐸 ∥ 𝐻𝑆 12.
13. 𝑂𝐸 + 𝐸𝑇 = 𝑂𝑇 13.
14. 𝑂𝐸 + 𝑂𝐸 = 𝑂𝑇 14.
15. 2𝑂𝐸 = 𝑂𝑇 15.
16. 𝐻𝑆 ≅ 𝑂𝑇 16.
17. 2𝑂𝐸 = 𝐻𝑆 17.
18. 𝑶𝑬 =
𝟏
𝟐
𝑯𝑺 (The segment
joining the midpoints of two
18.
Transitive Property
Definition of
Parallelogram
𝑂𝐸 is in the side of OT of HOTS
Segment Addition Postulate
Substitution (SN2)
Addition
Parallelogram Property
Substitution
Substitution
(SN 14 and 15)

14. For verifying
1. The group shall draw and cut a different kind of triangle out
of bond paper.
2. Choose a third side of a triangle. Mark each point of the
other two sides then connects the midpoints to form a
segment
Does the segment drawn look parallel to the third
side you chose?
Yes, it is parallel to third side of a triangle.

15. 3. Measure the segment drawn and the third side you chose.
Compare the lengths of the segments drawn and the
third side you chose. What did you observe?
4. Cut the triangle along the segment drawn.
What figures are formed after cutting the triangle
along the segment drawn?
The length of the segment drawn is one -half the length
of the third side.
The figures formed is triangle and a trapezoid.

16. 5. Reconnect the triangle with the other figure in such away
that their common vertex was a midpoint and that congruent
segments formed by a midpoint coincide.
*What new figure is formed?
*Make a conjecture to justify the new figure formed
after doing the activity. Explain your answer
A new figure formed is a parallelogram.
A conjecture can be made that a parallelogram is formed
when a segment joining the midpoints of any two sides is
cut and reconnecting the figures formed in such a way
that their common vertex was a midpoint and that
congruent segments formed by a midpoint coincide.

17. What can you say about your findings in
relation to those of your classmates?
Do you think that the findings apply to all
kinds of triangles? Why?
The findings are the same.
Yes, the findings can be applied to all kinds of
triangles because all figures formed are all
parallelograms.

18. A. Based on the results,
when can we have a
midline on a triangle?
When joining the
midpoints of the two
sides of a triangle.

19. B. How is the midline related with
the first 2 sides of a triangle?
The midline divides
the two sides of the
triangle into two
congruent segments.

20. C. How is the midline related to
the third side of the triangle?
The midline is
parallel to third
side and half as
long.

21. A. What did you learn in
our lesson?
The Midline
Theorem.

22. B. What is the midline
theorem on triangle?
The segment that joins
the midpoints of two
sides of a triangle is
parallel to the third
side and half as long.

23. C. How did we prove the
midline theorem on triangle?
By applying the
properties, theorems
on triangles and
using the different
properties of
equality.

24. 1. Given: ∆𝐴𝐵𝐶 , M and N are the
midpoints of 𝐴𝐶 and 𝐴𝐵,
respectively.
a. if MN = 12 cm,
then BC =
b. if AB = 36 cm,
then AN =
c. if MC = 12.5 cm,
then AC =
d. if AB = 2x,
then NB =
A
C B
M N
24 cm
18 cm
25 cm
x

25. 2. In the figure, S, H and E are the
midpoints of 𝑌𝑂,
𝑌𝑈, 𝑎𝑛𝑑 𝑈𝑂, respectively. Complete
each statement using the Midline
Theorem. How will you deal with a
triangle with more than 1 midline?
a. if SE = 12 then YU =
b. if OU = 34 then SH =
c. if HE = 16 then YO =
d. 𝑆𝐸 ∥
e. 𝑆𝐻 ∥
f. 𝐻𝐸 ∥
Y
U
H
S
E
O
24
17
32
𝒀𝑼
𝑶𝑼
𝑶𝒀

26. 3. Given the figure on the right, find the ff:
a. 𝐴𝐵 = 𝑥 + 15, 𝑋𝑌 = 4𝑥 − 12
Find x, AB and XY
a. Using the midline theorem
𝑨𝑩 =
𝟏
𝟐
𝑯𝒀
𝑥 + 15 =
4𝑥 − 12
2
2 𝑥 + 15 = 4𝑥 − 12
2𝑥 + 30 = 4𝑥 − 12
30 + 12 = 4𝑥 − 2𝑥
42 = 2𝑥
𝒙 = 𝟐𝟏
𝐴𝐵 = 𝑥 + 15
𝐴𝐵 = 21 + 15
𝑨𝑩 = 𝟑𝟔
𝑋𝑌 = 4𝑥 − 12
𝑋𝑌 = 4(21) − 12
𝑿𝒀 = 72
W
Y
X
A B

27. b. if 𝑊𝑋 = 𝑥 + 20 and 𝑊𝐴 = 3𝑥 + 5
Find AX
b. 𝑊𝑋 = 2(𝑊𝐴)
𝑊𝑋 = 2(3𝑥 + 5)
𝑥 + 20 = 6𝑥 + 10
20 − 10 = 6𝑥 − 𝑥
10 = 5𝑥
𝒙 = 𝟐
𝑾𝑨 = 𝑨𝑿
𝑊𝐴 = 3𝑥 + 5
𝑊𝐴 = 3(2) + 5
𝑊𝐴 = 6 + 5
𝑾𝑨 = 𝟏𝟏, therefore

28. Do you know that the midline theorem can also be
applied in our daily life?
Can you site examples of situation on how can it
be applied?
In Carpentry, Midline Theorem is applied in
Architecture when building and when making
blueprints for the buildings and houses such as roof
trusses, bridges and the like.

29. ROOF TRUSS DESIGN BALAY NEGRENSE
SAN CARLOS CITY BORROMEO
CATHEDRAL
LOVE BRIDGE OF KABANKALAN
CITY

30. Given: ∆𝑴𝑪𝑮 with A and I as midpoints of
𝑴𝑮, and 𝑪𝑮, respectively. Answer the following using the
given figure below. M
C
G
I
A
1. If AI = 10.5 cm
then MC = ___
2. If CG = 32 cm
then GI = ___
3. If AG = 7 cm and
CI = 8 cm then
what is MG + GC?
4. If 𝐴𝐼 = 3𝑥 − 2 and
𝑀𝐶 = 9𝑥 − 13 then what is the value
of x? length of 𝐴𝐼? Length of 𝑀𝐶?
What is the sum of AI + MC?
5. Given: 𝑀𝐺 ≅ 𝐶𝐺, 𝐴𝐺 = 2𝑦 −
1,𝐼𝐶 = 𝑦 + 5. What is the value of y?
How long are 𝑀𝐺 and 𝐶𝐺?

31. ASSIGNMENT
Directions: Find the value of the indicated variable using
the midline theorem.
9y
5y+3
6-2x
3x-4
6x-2
3x+5
Y+12 5y-3