# 5.1 Midsegment Theorem and Coordinate Proof

## Transcription

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. !
!!!!
π·πΈ β₯ ________ and π·πΈ = ! ________
Example 1: Use the Midsegment Theorem 1. !!!!
π·πΉ β₯ ________
2. !!!!
πΈπΉ β₯ ________
3. !!!!
π·πΈ β₯ ________
4. π΄π΅ = ________
5. π·πΈ = ________
6. π΄πΆ = ________
Example 2: Use βπ»πΊπΌ where πΏ, πΎ, πππ π½ are midpoints 7. Find πΊπΌ.
8. Find πΎπ½.
Geometry Chapter 5: Relationships within Triangles Use βπ΄π΅πΆ for problems 9 β 11, where D, E, and F are midpoints of the sides. 9. If π·πΈ = 2π₯ + 6 πππ πΆπ΅ = 6π₯ + 4, π€βππ‘ ππ  π·πΈ? 10. If π·πΉ = 3π₯ + 3 πππ π΄π΅ = 8π₯ β 4, π€βππ‘ ππ  π΄πΈ? 11. If π΄πΆ = 4π₯ β 2 πππ πΈπΉ = π₯ + 6, π€βππ‘ ππ  π΄πΆ? vοΆ A ___________________________________ proof involves placing geometric figures in a coordinate plane. 12. Graph βπ΄π΅πΆ with coordinates π΄ β1, 4 , π΅ 3, 0 , πΆ(5, 2). π 2, 3 is the midpoint of π΄πΆ. π(1, 2) is the midpoint of π΄π΅. a) Prove that ππ is parallel to π΅πΆ. (Use the slope formula.) b) Prove that ππ is half as long as π΅πΆ. (Use the distance formula.) Geometry Chapter 5: Relationships within Triangles 13. Show that the midsegment πΉπΊ is parallel to π΅πΆ and half as long. βπ΄π΅πΆ has vertices π΄ β3, 4 , π΅ β3, β2 , πΆ(5, 4). a) Find the coordinates of F, the midpoint of π΄π΅. (Use the midpoint formula) b) Find the coordinates of G, the midpoint of π΄πΆ. (Use the midpoint formula) c) Verify that πΉπΊ is parallel to π΅πΆ. (Use the slope formula to find the slopes of πΉπΊ πππ π΅πΆ) d) Verify that πΉπΊ is half as long as π΅πΆ. (Use the distance formula to find the lengths of πΉπΊ πππ π΅πΆ) 5.1 Homework: Use the diagram of βπ΄π΅πΆ where π·, πΈ, πππ πΉ are the midpoints of the sides. 1. π·πΈ β₯ _________ 2. πΉπΈ β₯ __________ 3. If π΄π΅ = 26, then π·πΉ = __________ 4. If πΆπΉ = 9, then π·πΈ = __________ 5. If πΈπΉ = 7, then π΄πΆ = __________ Geometry Chapter 5: Relationships within Triangles In the diagram of βπ½πΎπ where π, π, πππ π are the midpoints of the sides and ππΎ = 6, πΎπ = 8, πππ π½πΎ β₯ πΎπΏ. 6. Find the length of ππ. (Pythagorean Theorem) 7. Find the length of π½πΎ. 8. Find the length of ππ. 9. Find the perimeter of βπ½πΎπΏ. 10. Name all of the right angles in the diagram. Use the diagram of βπππ where π, π, πππ π are the midpoints of the sides. 11. If ππ = 3π₯ + 1 and ππ = 10π₯ β 6 then ππ = __________. 12. If ππ = π₯ β 1, and ππ = 3π₯ β 7, then ππ = __________. 13. If πβ πππ = 48°, then πβ πππ = __________. 14. If πβ πππ = 37°, then πβ πππ = __________. 15. Name a triangle that appears to be congruent to βπππ. ```