Midline Theorem-3rd Quarter lesson

1. MIDLINE (MID-SEGMENT)
THEOREM
QUARTER 3:
WEEK 3
9TH GRADE

2. MIDLINE THEOREM 9TH GRADE
The segment whose endpoints are
the midpoints of two sides of a
triangle is parallel to the third side
and half as long.
• In ∆𝑩𝑨𝑪, D and E are midpoints
of 𝑩𝑨 and 𝑨𝑪.
𝑩𝑫 ≅ 𝑫𝑨 and 𝑨𝑬 ≅ 𝑬𝑪
• By Vertical Angle Theorem
∆𝑨𝑬𝑫 ≅ ∆𝑭𝑬𝑪
• By SAS Postulate, ∆𝑨𝑬𝑫 ≅ ∆𝑨𝑬𝑭
wherer
𝑫𝑨 ≅ 𝑭𝑪 by CPCTC
𝑩𝑫 ≅ 𝑭𝑪 by Transitivity
∠𝑨𝑫𝑬 ≅ ∠𝑭𝑪𝑬 by CPCTC

3. MIDLINE THEOREM 9TH GRADE
The segment whose endpoints are
the midpoints of two sides of a
triangle is parallel to the third side
and half as long.
• Since a pair of alternate angles
are formed which are congruent,
then and then formed
parallelogram BDFC.
𝑩𝑫 ∥ 𝑭𝑪
• In a parallelogram BDFC, there
are two pairs of opposite sides
that are congruent 𝑫𝑭 ∥ 𝑩𝑪 and
𝑫𝑭 ≅ 𝑩𝑪 , likewise 𝑫𝑬 ∥ 𝑩𝑪
• E is the midpoint of 𝑫𝑭,
therefore 𝑫𝑬 ≅ 𝑬𝑭 and 𝑫𝑭 ≅
𝑩𝑪

4. MIDLINE THEOREM 9TH GRADE
The segment whose endpoints are
the midpoints of two sides of a
triangle is parallel to the third side
and half as long.
• E is the midpoint of 𝑫𝑭,
therefore 𝑫𝑬 ≅ 𝑬𝑭 and 𝑫𝑭 ≅
𝑩𝑪
𝑫𝑬 + 𝑬𝑭 = 𝑫𝑭
𝟐𝑫𝑬 = 𝑩𝑪
𝑫𝑬 =
𝟏
𝟐
𝑩𝑪

5. 9TH GRADE
EXAMPLE 1
In ∆𝐵𝐸𝐴, X and Y are midpoints of 𝐵𝐸 and 𝐸𝐴 respectively
Complete the following statements:
1. 𝑩𝑿 ≅ ______
2. 𝑩𝑨 ≅ 𝟐 ( _____)
3. 𝑬𝑨 ≅ 𝑬𝒀 + _____
4. If 𝑩𝑨 = 𝟑𝟔, 𝒕𝒉𝒆𝒏 𝑿𝒀 = _____
5. If 𝑬𝑿 = 𝟏𝟎, 𝒕𝒉𝒆𝒏 𝑩𝑿 = _____
𝑬𝑿
𝑿𝒀
𝒀𝑨
𝟏𝟖
𝟏𝟎

6. 9TH GRADE
EXAMPLE 2
Find the missing length as indicated.
a. Find x.
b. Find 𝑳𝑵.
c. Find x.
d. Find 𝑺𝑹.

7. 9TH GRADE
EXAMPLE 3
Given: ∆𝐴𝐶𝐸, with B and D as midpoints of 𝐶𝐴 and 𝐶𝐸
respectively.
A. If 𝑪𝑫 = 𝟏𝟗, 𝒕𝒉𝒆𝒏
𝟏
𝟐
𝑪𝑬 = _____
B. If 𝑩𝑫 = 𝟐𝟏, 𝒕𝒉𝒆𝒏 𝑩𝑫 + 𝑨𝑬 = _____
C. If 𝑩𝑫 = 𝟐𝒙 − 𝟏, 𝑨𝑬 = 𝒙 + 𝟒,
𝒕𝒉𝒆𝒏 𝑩𝑫 = _____
D. If 𝑩𝑪 = 𝟐𝒂 − 𝟏, 𝑩𝑨 = 𝟒𝒂 − 𝟏𝟕,
𝒕𝒉𝒆𝒏 𝑩𝑨 = _____
𝟏𝟗
𝟔𝟑
𝟑
𝟏𝟓

8. 9TH GRADE
EXAMPLE 4
∥
C
E
A
V
R
∥
−𝟐𝒙 − 𝟔
𝒙 + 𝟏𝟓
𝒙 + 𝟐𝟏

9. 9TH GRADE
MIDLINE OR MID-SEGMENT
- Is a segment connecting the midpoints of any two sides of a
triangle.
G
M
A
MIDPOINT
Is the middle point
of a line segment
I C
𝑴𝑰 ≅ 𝑰𝑮 𝑴𝑪 ≅ 𝑪𝑨
𝑰𝑪 is a midline of ∆𝑴𝑨𝑮

10. MIDLINE THEOREM 9TH GRADE
The midline of a triangle is parallel to and is half
the length of the 3rd side.
𝑰𝑪 ∥ 𝑮𝑨
𝑰𝑪 is a midline of ∆𝑴𝑨𝑮
G
M
A
I C
𝑰𝑪 =
𝑮𝑨
𝟐

11. 9TH GRADE
EXAMPLE 1
In the figure below, 𝐷𝐸 is midline of ∆𝐴𝐵𝐶. If 𝐷𝐸 is
parallel to 𝐴𝐶, and 𝐴𝐶 has a length of 30 feet, how long
is 𝐷𝐸?
B
D E
A C

12. 9TH GRADE
EXAMPLE 2
In the triangle below, 𝑫 is the midpoint of 𝐴𝐵 and E is
the midpoint of 𝐵𝐶, find the measure of 𝐴𝐶 if 𝐷𝐸 = 7.
B
D E
A C

13. 9TH GRADE
EXAMPLE 3
A and B are midpoints of 𝑋𝑌 and 𝑍𝑌 as shown in the
figure below. If 𝑋𝑍 = 34 and 𝐴𝐵 = 3x − 1, solve for x.
Y
A B
X Z
𝟑𝐱 − 𝟏
𝟑𝟒

14. 9TH GRADE
EXAMPLE 4
Determine the following:
a. The value of x
b. The length of the base 𝐿𝑁.
M
E
L
F
N
𝒙 + 𝟐
𝒙 + 𝟏𝟎

15. 9TH GRADE
ACTIVITY
Answer the following.