5.1 Midsegment Theorem and Coordinate Proof

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5.1 Midsegment Theorem and Coordinate Proof
Geometry Chapter 5: Relationships within Triangles 5.1-­β€ Midsegment Theorem and Coordinate Proof SWBAT: use properties of midsegments and write coordinate proofs. Common Core: G.CO.10
Vocabulary A ______________________________ of a triangle is a segment that connects the midpoints of two sides of the triangle. Every triangle has three midsegments. Theorem 5.1-­β€ Midsegment Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side. !
!!!!
𝐷𝐸 βˆ₯ ________ and 𝐷𝐸 = ! ________
Example 1: Use the Midsegment Theorem 1. !!!!
𝐷𝐹 βˆ₯ ________
2. !!!!
𝐸𝐹 βˆ₯ ________
3. !!!!
𝐷𝐸 βˆ₯ ________
4. 𝐴𝐡 = ________
5. 𝐷𝐸 = ________
6. 𝐴𝐢 = ________
Example 2: Use βˆ†π»πΊπΌ where 𝐿, 𝐾, π‘Žπ‘›π‘‘ 𝐽 are midpoints 7. Find 𝐺𝐼.
8. Find 𝐾𝐽.
Geometry Chapter 5: Relationships within Triangles Use βˆ†π΄π΅πΆ for problems 9 – 11, where D, E, and F are midpoints of the sides. 9. If 𝐷𝐸 = 2π‘₯ + 6 π‘Žπ‘›π‘‘ 𝐢𝐡 = 6π‘₯ + 4, π‘€β„Žπ‘Žπ‘‘ 𝑖𝑠 𝐷𝐸? 10. If 𝐷𝐹 = 3π‘₯ + 3 π‘Žπ‘›π‘‘ 𝐴𝐡 = 8π‘₯ βˆ’ 4, π‘€β„Žπ‘Žπ‘‘ 𝑖𝑠 𝐴𝐸? 11. If 𝐴𝐢 = 4π‘₯ βˆ’ 2 π‘Žπ‘›π‘‘ 𝐸𝐹 = π‘₯ + 6, π‘€β„Žπ‘Žπ‘‘ 𝑖𝑠 𝐴𝐢? v A ___________________________________ proof involves placing geometric figures in a coordinate plane. 12. Graph βˆ†π΄π΅πΆ with coordinates 𝐴 βˆ’1, 4 , 𝐡 3, 0 , 𝐢(5, 2). 𝑃 2, 3 is the midpoint of 𝐴𝐢. 𝑄(1, 2) is the midpoint of 𝐴𝐡. a) Prove that 𝑃𝑄 is parallel to 𝐡𝐢. (Use the slope formula.) b) Prove that 𝑃𝑄 is half as long as 𝐡𝐢. (Use the distance formula.) Geometry Chapter 5: Relationships within Triangles 13. Show that the midsegment 𝐹𝐺 is parallel to 𝐡𝐢 and half as long. βˆ†π΄π΅πΆ has vertices 𝐴 βˆ’3, 4 , 𝐡 βˆ’3, βˆ’2 , 𝐢(5, 4). a) Find the coordinates of F, the midpoint of 𝐴𝐡. (Use the midpoint formula) b) Find the coordinates of G, the midpoint of 𝐴𝐢. (Use the midpoint formula) c) Verify that 𝐹𝐺 is parallel to 𝐡𝐢. (Use the slope formula to find the slopes of 𝐹𝐺 π‘Žπ‘›π‘‘ 𝐡𝐢) d) Verify that 𝐹𝐺 is half as long as 𝐡𝐢. (Use the distance formula to find the lengths of 𝐹𝐺 π‘Žπ‘›π‘‘ 𝐡𝐢) 5.1 Homework: Use the diagram of βˆ†π΄π΅πΆ where 𝐷, 𝐸, π‘Žπ‘›π‘‘ 𝐹 are the midpoints of the sides. 1. 𝐷𝐸 βˆ₯ _________ 2. 𝐹𝐸 βˆ₯ __________ 3. If 𝐴𝐡 = 26, then 𝐷𝐹 = __________ 4. If 𝐢𝐹 = 9, then 𝐷𝐸 = __________ 5. If 𝐸𝐹 = 7, then 𝐴𝐢 = __________ Geometry Chapter 5: Relationships within Triangles In the diagram of βˆ†π½πΎπ‘™ where 𝑅, 𝑆, π‘Žπ‘›π‘‘ 𝑇 are the midpoints of the sides and 𝑅𝐾 = 6, 𝐾𝑆 = 8, π‘Žπ‘›π‘‘ 𝐽𝐾 βŠ₯ 𝐾𝐿. 6. Find the length of 𝑅𝑆. (Pythagorean Theorem) 7. Find the length of 𝐽𝐾. 8. Find the length of 𝑅𝑇. 9. Find the perimeter of βˆ†π½πΎπΏ. 10. Name all of the right angles in the diagram. Use the diagram of βˆ†π‘€π‘π‘‚ where 𝑋, π‘Œ, π‘Žπ‘›π‘‘ 𝑍 are the midpoints of the sides. 11. If π‘Œπ‘ = 3π‘₯ + 1 and 𝑀𝑁 = 10π‘₯ βˆ’ 6 then π‘Œπ‘ = __________. 12. If π‘Œπ‘‹ = π‘₯ βˆ’ 1, and 𝑀𝑂 = 3π‘₯ βˆ’ 7, then 𝑀𝑂 = __________. 13. If π‘šβˆ π‘€π‘‚π‘ = 48°, then π‘šβˆ π‘€π‘π‘‹ = __________. 14. If π‘šβˆ π‘€π‘‹π‘ = 37°, then π‘šβˆ π‘€π‘π‘‚ = __________. 15. Name a triangle that appears to be congruent to βˆ†π‘π‘‚π‘Œ.