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Exploring Midsegments in Triangles: Lengths and Proportionality

This content delves into midsegments of triangles, which connect the midpoints of two sides. Utilizing the Triangle Proportionality Theorem, we explore how to find segment lengths through substitutive equations and the Cross Product Property. Examples demonstrate how to solve for unknown values using segment addition and proportional relationships. Learn to determine if segments are parallel, apply the Midsegment Theorem for lengths, and practice with exercises to enhance your understanding of triangle geometry.

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Exploring Midsegments in Triangles: Lengths and Proportionality

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  1. Bell Ringer

  2. Triangle Mid-segments

  3. Midsegment of a triangle • A Midsegmentof a triangle is the segments that connects the midpoints to two sides of a triangle.

  4. Example 1 Find Segment Lengths Find the value of x. SOLUTION Triangle Proportionality Theorem = Substitute 4 for CD, 8 for DB, x for CE, and 12 for EA. x = 12 4·12=8 ·x Cross product property 4 48 = 8x 8 Multiply. 8x 48 CD CE = Divide each side by 8. 8 8 DB EA 6 = x Simplify.

  5. Example 2 Find Segment Lengths Find the value of y. SOLUTION You know that PS = 20 and PT = y. By the Segment Addition Postulate, TS = 20 – y. = Triangle Proportionality Theorem y = 20 – y 3 Substitute 3 for PQ, 9 for QR, y for PT, and (20 – y) for TS. 9 PQ PT Cross product property 3(20 – y)=9 ·y TS QR 60 – 3y = 9y Distributive property

  6. Example 2 Find Segment Lengths 60 – 3y + 3y = 9y + 3y Add 3y to each side. 60 = 12y Simplify. Divide each side by 12. 5 = y Simplify. 12y 60 = 12 12

  7. Example 3 SOLUTION Find and simplify the ratios of the two sides divided by MN. , MN is not parallel to GH. Because ANSWER Determine Parallels Given the diagram, determine whether MN is parallel to GH. = = = = LM LN 8 3 8 3 48 56 3 1 ≠ 1 3 21 16 MG NH

  8. Find Segment Lengths and Determine Parallels Now You Try  Find the value of the variable. 1. 8 10 ANSWER ANSWER 2.

  9. Find Segment Lengths and Determine Parallels Given the diagram, determine whether QR is parallel to ST. Explain. ANSWER || Yes; = so QR ST by the Converse of the Triangle Proportionality Theorem. Now You Try  3. ≠ ANSWER no; 4. 4 6 17 15 12 8 23 21

  10. Example 4 Use the Midsegment Theorem Find the length of QS. SOLUTION From the marks on the diagram, you know S is the midpoint of RT, and Q is the midpoint of RP. Therefore, QS is a midsegment of PRT. Use the Midsegment Theorem to write the following equation. QS= PT = (10) = 5 1 1 2 2 The length of QS is 5. ANSWER

  11. Use the Midsegment Theorem Now You Try  Find the value of the variable. 5. 6. 24 8 28 ANSWER ANSWER ANSWER 7. Use the Midsegment Theorem to find the perimeter of ABC.

  12. Now You Try 

  13. Now You Try 

  14. Now You Try 

  15. Now You Try 

  16. Page 390

  17. Complete Pages 390-392 #s 10-28 even only

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