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5.1 Midsegment Theorem and Coordinate Proof

5.1 Midsegment Theorem and Coordinate Proof. Essential Question:. How do you write a coordinate proof?. You will use properties of midsegments and write coordinate proofs.

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5.1 Midsegment Theorem and Coordinate Proof

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  1. 5.1 Midsegment Theorem and Coordinate Proof Essential Question: How do you write a coordinate proof? • You will use properties of midsegments and write coordinate proofs. You will learn how to answer this question by placing a figure in the coordinate plane, assigning coordinates to the vertices, and then using the midpoint, distance, and/or slope formulas.

  2. Triangles are used for strength in roof trusses. In the diagram, UVand VWare midsegments of RST. Find UVand RS. 1 1 2 2 RT ( 90 in.) = 45 in. = = UV VW ( 57 in.) 2 2 = 114 in. = = RS EXAMPLE 1 Use the Midsegment Theorem to find lengths CONSTRUCTION SOLUTION

  3. ANSWER UW 2. In Example 1, suppose the distance UWis 81 inches. Find VS. ANSWER 81 in. for Example 1 GUIDED PRACTICE 1. Copy the diagram in Example 1. Draw and name the third midsegment.

  4. In the kaleidoscope image, AEBEand AD CD. Show that CB DE. Because AE BEand AD CD , E is the midpoint of ABand Dis the midpoint of ACby definition. Then DEis a midsegment of ABCby definition and CB DEby the Midsegment Theorem. EXAMPLE 2 Use the Midsegment Theorem SOLUTION

  5. A rectangle A scalene triangle a. b. SOLUTION It is easy to find lengths of horizontal and vertical segments and distances from (0, 0), so place one vertex at the origin and one or more sides on an axis. EXAMPLE 3 Place a figure in a coordinate plane Place each figure in a coordinate plane in a way that is convenient for finding side lengths. Assign coordinates to each vertex.

  6. b. a. Notice that you need to use three different variables. Let hrepresent the length and krepresent the width. EXAMPLE 3 Place a figure in a coordinate plane

  7. In Example 2, if Fis the midpoint of CB, what do you know about DF? DF is a midsegment of ABC. DF ABandDFis half the length of AB. 4. Show another way to place the rectangle in part (a) of Example 3 that is convenient for finding side lengths. Assign new coordinates. ANSWER for Examples 2 and 3 GUIDED PRACTICE 3. ANSWER

  8. ANSWER Yes; the length of one side is d. 6. A square has vertices (0, 0), (m, 0), and (0, m). Find the fourth vertex. ANSWER (m, m) for Examples 2 and 3 GUIDED PRACTICE 5. Is it possible to find any of the side lengths in part (b) of Example 3 without using the Distance Formula? Explain.

  9. Place an isosceles right triangle in a coordinate plane. Then find the length of the hypotenuse and the coordinates of its midpoint M. SOLUTION Place PQOwith the right angle at the origin. Let the length of the legs be k. Then the vertices are located at P(0, k), Q(k, 0), and O(0, 0). EXAMPLE 4 Apply variable coordinates

  10. = = = = k 2 k k 2 2 = 0 + k , k +0 M( ) M( , ) 2 2 2 2 2 2 2 2 2 k + (– k) 2k k + k (k–0) + (0–k) EXAMPLE 4 Apply variable coordinates Use the Distance Formula to find PQ. PQ = Use the Midpoint Formula to find the midpoint Mof the hypotenuse.

  11. Write a coordinate proof of the Midsegment Theorem for one midsegment. E(q+p, r) D(q, r) = = GIVEN : PROVE : SOLUTION 1 DE is a midsegment of OBC. 2 DE OCand DE = OC STEP1 Place OBCand assign coordinates. Because you are finding midpoints, use 2p, 2q, and 2r. Then find the coordinates of Dand E. 2q + 2p, 2r + 0 2q + 0, 2r + 0 E( ) D( ) 2 2 2 2 EXAMPLE 5 Prove the Midsegment Theorem

  12. STEP 2 Prove DE OC. The y-coordinates of Dand Eare the same, so DEhas a slope of 0. OCis on the x-axis, so its slope is 0. 1 Because their slopes are the same, DE OC. 2 Prove DE = OC. Use the Ruler Postulate to find DEand OC. STEP3 = 2p 2p – 0 (q +p) – q OC= = p DE= So, the length of DEis half the length of OC EXAMPLE 5 Prove the Midsegment Theorem

  13. In Example 5, find the coordinates of F, the midpoint of OC. Then show that EF OB. ANSWER r 0 (p, 0); slope of EF = = , slope of OB = = , the slopes of EF and OB are both , making EF || OB. (q + p)  p 2r 0 2q 0 r r r q q q for Examples 4 and 5 GUIDED PRACTICE 7.

  14. Graph the points O(0, 0), H(m, n), and J(m, 0). Is OHJa right triangle? Find the side lengths and the coordinates of the midpoint of each side. ANSWER Sample: yes; OJ = m, JH = n, HO = m2 + n2, OJ: ( , 0), JH: (m, ), HO: ( , ) m n m n 2 2 2 2 for Examples 4 and 5 GUIDED PRACTICE 8.

  15. Use the figure below for Exercises 1–4. 26 10 34 in. ANSWER ANSWER ANSWER 3. If the perimeter of RST = 68 inches, find the Perimeter of UVW. Daily Homework Quiz 1. If UV = 13, find RT. 2. If ST = 20, find UW.

  16. 6 ANSWER ANSWER Daily Homework Quiz 4. If VW = 2x – 4, and RS = 3x – 3, what isVW? 5. Place a rectangle in a coordinate plane so its vertical side has length a and its horizontal side has width 2a. Label the coordinates of each vertex.

  17. Essential Question: How do you write a coordinate proof? • Assign coordinates to vertices that are convenient for finding lengths. • Use coordinates to find midpoints, distances, and slopes. Assign convenient coordinates to vertices and use the midpoint, distance, and slope formulas to generate the proof. • You will use properties of midsegments and write coordinate proofs.

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