Midsegments of Triangles Skill 25

Objective HSG-SRT.5: Students are responsible for using properties of Midsegments to solve problems.

Theorem 23: Triangle Midsegment Theorem If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and half as long. C E D A B If D is the midpoint of 𝑪𝑨 , and E is the midpoint of 𝑪𝑩 Then 𝑫𝑬 ∥ 𝑨𝑩 and 𝐃𝐄= 𝟏 𝟐 𝑨𝑩

Example 1; Identifying Parallel Segments What are three pairs of parallel segments in ∆𝐷𝐸𝐹? Since R is the midpoint of 𝑫𝑬 and S is the midpoint of 𝑬𝑭 D E F T S R 𝑹𝑺 ∥ 𝑫𝑭 Since R is the midpoint of 𝑫𝑬 and T is the midpoint of 𝑫𝑭 𝑹𝑻 ∥ 𝑬𝑭 Since S is the midpoint of 𝑬𝑭 and T is the midpoint of 𝑫𝑭 𝑺𝑻 ∥ 𝑬𝑫

Example 2; Finding Lengths In ∆𝑄𝑆𝑇 T, U, and B are midpoints. What are the lengths of 𝑇𝑈 , 𝑈𝐵 , and 𝑄𝑅 ? U B T S R Q 40 30 50 𝑻𝑼= 𝟏 𝟐 𝑺𝑹 𝑻𝑼= 𝟏 𝟐 𝟒𝟎 𝑻𝑼=𝟐𝟎 𝑼𝑩= 𝟏 𝟐 𝑸𝑺 𝑼𝑩= 𝟏 𝟐 𝟓𝟎 𝑼𝑩=𝟐𝟓 𝑻𝑩= 𝟏 𝟐 𝑸𝑹 𝟑𝟎 = 𝟏 𝟐 𝑸𝑹 𝟐 𝟑𝟎 = 𝑸𝑹 𝑼𝑩=𝟔𝟎

Example 3; Using the Midsegment of a Triangle A geologist wants to determine the distance, 𝐴𝐵, across a sinkhole. Choosing a point E outside the sinkhole, she finds the distance 𝐴𝐸 and 𝐵𝐸. She locates the midpoints C and D of 𝐴𝐸 and 𝐵𝐸 and then measures 𝐶𝐷 . What is the distance across the sinkhole? A B C D E She measures 𝑪𝑫=𝟒𝟔 𝒇𝒕. 𝑪𝑫= 𝟏 𝟐 𝑨𝑩 𝑨𝑩=𝟐 𝑪𝑫 𝑨𝑩=𝟐 𝟒𝟔 𝑨𝑩=𝟗𝟐 𝒇𝒆𝒆𝒕 46’

Example 4; Using the Midsegment of a Triangle 𝐶𝐷 is a bridge being built over a lake as shown in the figure at the left. What is the length of the bridge? Notice: 𝑪 is a midpoint Notice: 𝑫 is a midpoint 𝑪𝑫= 𝟏 𝟐 𝟐𝟔𝟒𝟎 𝑪𝑫=𝟏𝟑𝟐𝟎 𝒇𝒆𝒆𝒕 963 ft. 2640 ft. C D

#25: Midsegments in Triangles Questions? Summarize Notes Homework Video Quiz