Special Lines in Triangles - Medians

New York State Common Core Math Geometry, Module 1, Lesson 30

Worksheets for Geometry

Student Outcomes

• Students examine the relationships created by special lines in triangles, namely medians.

Special Lines in Triangles - Medians

Classwork

Opening Exercise

In △ 𝐴𝐵𝐶 to the right, 𝐷 is the midpoint of 𝐴𝐵, 𝐸 is the midpoint of 𝐵𝐶, and 𝐹 is the midpoint of 𝐴𝐶. Complete each statement below.

𝐷𝐸 is parallel to ____ and measures ____ the length of ____. 𝐷𝐹 is parallel to ____ and measures ____ the length of ____. 𝐸𝐹 is parallel to ____ and measures ____ the length of ____.

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Discussion

In the previous two lessons, we proved that (a) the midsegment of a triangle is parallel to the third side and half the length of the third side and (b) diagonals of a parallelogram bisect each other. We use both of these facts to prove the following assertion:

All medians of a triangle are ____. That is, the three medians of a triangle (the segments connecting each vertex to the midpoint of the opposite side) meet at a single point. This point of concurrency is called the ____, or the center of gravity, of the triangle. The proof also shows a length relationship for each median: The length from the vertex to the centroid is ____ the length from the centroid to the midpoint of the side.

Example 1

Provide a valid reason for each step in the proof below.

Given: △ 𝐴𝐵𝐶 with 𝐷, 𝐸, and 𝐹 the midpoints of sides ̅𝐴𝐵̅̅̅, 𝐵𝐶̅̅̅̅, and 𝐴𝐶̅̅̅̅, respectively
Prove: The three medians of △ 𝐴𝐵𝐶 meet at a single point.

(1) Draw midsegment 𝐷𝐸. Draw 𝐴𝐸 and 𝐷𝐶; label their intersection as point 𝐺.
(2) Construct and label the midpoint of 𝐴𝐺 as point 𝐻 and the midpoint of 𝐺𝐶 as point 𝐽.
(3) 𝐷𝐸 ∥ 𝐴𝐶,
(4) 𝐻𝐽 ∥ 𝐴𝐶,
(5) 𝐷𝐸 ∥ 𝐻𝐽,
(6) 𝐷𝐸 = 1/2 𝐴𝐶 and 𝐻𝐽 = 1/2𝐴𝐶,
(7) 𝐷𝐸𝐽𝐻 is a parallelogram.
(8) 𝐻𝐺 = 𝐸𝐺 and 𝐽𝐺 = 𝐷𝐺,
(9) 𝐴𝐻 = 𝐻𝐺 and 𝐶𝐽 = 𝐽𝐺,
(10) 𝐴𝐻 = 𝐻𝐺 = 𝐺𝐸 and 𝐶𝐽 = 𝐽𝐺 = 𝐺𝐷,
(11) 𝐴𝐺 = 2𝐺𝐸 and 𝐶𝐺 = 2𝐺𝐷,
(12) We can complete Steps (1)–(11) to include the median from 𝐵; the third median, 𝐵𝐹, passes through point 𝐺, which divides it into two segments such that the longer part is twice the shorter.
(13) The intersection point of the medians divides each median into two parts with lengths in a ratio of 2:1; therefore, all medians are concurrent at that point.

The three medians of a triangle are concurrent at the , or the center of gravity. This point of concurrency divides the length of each median in a ratio of ; the length from the vertex to the centroid is the length from the centroid to the midpoint of the side.

Example 2

In △ 𝐴𝐵𝐶, the medians are concurrent at 𝐹. 𝐷𝐹 = 4, 𝐵𝐹 = 16, 𝐴𝐺 = 30. Find each of the following measures.
a. 𝐹𝐶 =
b. 𝐷𝐶 =
c. 𝐴𝐹 =
d. 𝐵𝐸 =
e. 𝐹𝐺 =
f. 𝐸𝐹 =

Example 3

In the figure to the right, △ 𝐴𝐵𝐶 is reflected over 𝐴𝐵 to create △ 𝐴𝐵𝐷. Points 𝑃, 𝐸, and 𝐹 are midpoints of 𝐴𝐵, 𝐵𝐷, and 𝐵𝐶, respectively. If 𝐴𝐻 = 𝐴𝐺, prove that 𝑃𝐻 = 𝐺𝑃.

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