Midsegments of Triangles

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Midsegments of Triangles
Lesson 5-1 Check Skills You’ll Need (For help, go to Lesson 1-8 and page 165.) Find the coordinates of the midpoint of each segment. 1. AB with A(–2, 3) and B(4, 1) 2. CD with C(0, 5) and D(3, 6) 3. EF with E(–4, 6) and F(3, 10) 4. GH with G(7, 10) and H(–5, –8) Find the slope of the line containing each pair of points. 5. A(–2, 3) and B(3, 1) 6. C(0, 5) and D(3, 6) 7. E(–4, 6) and F(3, 10) 8. G(7, 10) and H(–5, –8) ( 1, 2) Check Skills You’ll Need 5-1

Midsegments of Triangles
Lesson 5-1 A midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, which form the midsegment triangle. 5-1

Midsegments of Triangles
Lesson 5-1 5-1

Midsegments of Triangles
Lesson 5-1 You have used coordinate geometry to find the midpoint of a line segment and to find the distance between two points. Coordinate geometry can also be used to prove conjectures. A coordinate proof is a style of proof that uses coordinate geometry and algebra. The first step of a coordinate proof is to position the given figure in the plane. You can use any position, but some strategies can make the steps of the proof simpler. 5-1

Midsegments of Triangles
Lesson 5-1 5-1

Midsegments of Triangles
Lesson 5-1 Once the figure is placed in the coordinate plane, you can use slope, the coordinates of the vertices, the Distance Formula, or the Midpoint Formula to prove statements about the figure. 5-1

To do this, assign variables as the coordinates of the vertices.
Midsegments of Triangles Lesson 5-1 A coordinate proof can also be used to prove that a certain relationship is always true. You can prove that a statement is true for all triangles without knowing the side lengths. To do this, assign variables as the coordinates of the vertices. 5-1

Midsegments of Triangles
Lesson 5-1 If a coordinate proof requires calculations with fractions, choose coordinates that make the calculations simpler. For example, use multiples of 2 when you are to find coordinates of a midpoint. Once you have assigned the coordinates of the vertices, the procedure for the proof is the same, except that your calculations will involve variables. 5-1

Midsegments of Triangles
Lesson 5-1 Do not use both axes when positioning a figure unless you know the figure has a right angle. Caution! Because the x- and y-axes intersect at right angles, they can be used to form the sides of a right triangle. Remember! 5-1

Proof of the Triangle Midsegment Theorem
Midsegments of Triangles Lesson 5-1 Proof of the Triangle Midsegment Theorem 5-1

Midsegments of Triangles
Lesson 5-1 Additional Examples Finding Lengths In XYZ, M, N, and P are midpoints. The perimeter of MNP is 60. Find NP and YZ. Because the perimeter of MNP is 60, you can find NP. NP + MN + MP = 60 Definition of perimeter NP = 60 Substitute 24 for MN and 22 for MP. NP + 46 = 60 Simplify. NP = 14 Subtract 46 from each side. Use the Triangle Midsegment Theorem to find YZ. MP = YZ Triangle Midsegment Theorem 22 = YZ Substitute 22 for MP. 44 = YZ Multiply each side by 2. 1 2 Quick Check 5-1

Midsegments of Triangles
Lesson 5-1 Additional Examples Finding Angle Measures Find m AMN and m ANM. MN and BC are cut by transversal AB , so AMN and B are corresponding angles. MN || BC by the Triangle Midsegment Theorem, so AMN B because parallel lines cut by a transversal form congruent corresponding angles. m AMN = 75 because congruent angles have the same measure. In AMN, AM = AN, so m ANM = m AMN by the Isosceles Triangle Theorem. m ANM = 75 by substituting 75 for m AMN. Quick Check 5-1

Midsegments of Triangles
Lesson 5-1 Additional Examples Real-World Connection Explain why Dean could use the Triangle Midsegment Theorem to measure the length of the lake. Solution: To find the length of the lake, Dean starts at the end of the lake and paces straight along that end of the lake. He counts the number of strides (35). Where the lake ends, he sets a stake. He paces the same number of strides (35) in the same direction and sets another stake. The first stake marks the midpoint of one side of a triangle. Dean paces from the second stake straight to the other end of the lake. He counts the number of his strides (236). 5-1

Midsegments of Triangles
Lesson 5-1 Additional Examples (continued) Dean finds the midsegment of the second side by pacing exactly half the number of strides back toward the second stake. He paces 118 strides. From this midpoint of the second side of the triangle, he returns to the midpoint of the first side, counting the number of strides (128). Dean has paced a triangle. He has also formed a midsegment of a triangle whose third side is the length of the lake. By the Triangle Midsegment Theorem, the segment connecting the two midpoints is half the distance across the lake. So, Dean multiplies the length of the midsegment by 2 to find the length of the lake. Quick Check 5-1

Midsegments of Triangles
Lesson 5-1 Lesson Quiz In GHI, R, S, and T are midpoints. 1. Name all the pairs of parallel segments. 2. If GH = 20 and HI = 18, find RT. 3. If RH = 7 and RS = 5, find ST. 4. If m G = 60 and m I = 70, find m GTR. 5. If m H = 50 and m I = 66, find m ITS. 6. If m G = m H = m I and RT = 15, find the perimeter of GHI. RT || HI, RS || GI, ST || HG 9 7 70 64 90 5-1

Midsegments of Triangles
Lesson 5-1 Check Skills You’ll Need Solutions x1 + x2 2 y1 + y2 2 –2 + 4 2 3 + 1 2 2 4 2 , = , = , = ( 1, 2) , = , = , , = , = , = (– , 8) , = , = , = (1,1) 5. m = = = = – 6. m = = = 7. m = = = 8. m = = = = x1 + x2 2 y1 + y2 2 0 + 3 2 5 + 6 2 3 2 11 2 x1 + x2 2 y1 + y2 2 –4 + 3 2 6 + 10 2 –1 2 16 2 1 2 x1 + x2 2 y1 + y2 2 7 + (–5) 2 10 + (–8) 2 2 2 y2 – y1 x2 – x1 1 – 3 3 – (–2) –2 5 2 5 y2 – y1 x2 – x1 6 – 5 3 – 0 1 3 y2 – y1 x2 – x1 10 – 6 3 – (–4) 4 7 y2 – y1 x2 – x1 –8 – 10 –5 – 7 – 18 – 12 3 2 5-1