5.1 midsegment theorem and coordinate proof
- 1. 5.1 Midsegment Theorem and Coordinate Proof 5.1 Bell Thinger In Exercises 1– 4, use A(0, 10), B(24, 0), and C(0, 0). 1. Find AB. ANSWER 26 2. Find the midpoint of CA. ANSWER (0, 5) 3. Find the midpoint of AB. ANSWER (12, 5) 4. Find the slope of AB. – 5 ANSWER 12
- 2. 5.1
- 3. Example 1 5.1 CONSTRUCTION Triangles are used for strength in roof trusses. In the diagram, UV and VW are midsegments of RST. Find UV and RS. SOLUTION 1 = 2 UV . RT 1 = 2 ( 90 in.) = 45 in. RS = 2 . VW = 2 ( 57 in.) = 114 in.
- 4. Guided Practice 5.1 1. UV and VW are midsegments. Name the third midsegment. ANSWER UW 2. Suppose the distance UW is 81 inches. Find VS. ANSWER 81 in.
- 5. Example 2 5.1 In the kaleidoscope image, AE ≅ BE and AD ≅ CD . Show that CB DE . SOLUTION Because AE ≅ BE and AD ≅CD , E is the midpoint of AB and D is the midpoint of AC by definition. Then DE is a midsegment of ABC by definition and CB DE by the Midsegment Theorem.
- 6. 5.1
- 7. Example 3 5.1 Place each figure in a coordinate plane in a way that is convenient for finding side lengths. Assign coordinates to each vertex. a. Rectangle: length is h and width is k b. Scalene triangle: one side length is d 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.
- 8. Example 3 5.1 a. Let h represent the length and k represent the width. b. Notice that you need to use three different variables.
- 9. Guided Practice 5.1 3. Is it possible to find any of the side lengths without using the Distance Formula? Explain. ANSWER Yes; the length of one side is d. 4. A square has vertices (0, 0), (m, 0), and (0, m). Find the fourth vertex. ANSWER (m, m)
- 10. Example 4 5.1 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 PQO with 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).
- 11. Example 4 5.1 Use the Distance Formula to find PQ. PQ = 2 2 (k – 0) + (0 – k) = = 2 2 k + (– k) = 2 2k = k 2 k +k 2 2 Use the Midpoint Formula to find the midpoint M of the hypotenuse. M( 0 + k , k + 0 ) = M( k , k ) 2 2 2 2
- 12. Guided Practice 5.1 5. Graph the points O(0, 0), H(m, n), and J(m, 0). Is OHJ a 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, m n OJ: ( 2 , 0), JH: (m, 2 ), HO: ( m , n ) 2 2
- 13. Exit Slip 5.1 If UV = 13, find RT. ANSWER 26 1. 2. If ST = 20, find UW. ANSWER 10 3. If the perimeter of RST = 68 inches, find the perimeter of UVW. ANSWER 34 in. 4. If VW = 2x – 4, and RS = 3x – 3, what is VW? ANSWER 6
- 14. Exit Slip 5.1 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. ANSWER
- 15. 5.1 Homework Pg 312-315 #27, 28, 29, 30, 35