1 / 20

Warm Up

Midsegment Theorem and Coordinate Proof. Warm Up. Lesson Presentation. Lesson Quiz. 26. ANSWER. 2. Find the midpoint of CA. (0, 5). ANSWER. Warm-Up. In Exercises 1– 4, use A (0, 10), B (24, 0), and C (0, 0). 1. Find AB. 3. Find the midpoint of AB. (12, 5). ANSWER.

hank
Download Presentation

Warm Up

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Midsegment Theorem and Coordinate Proof Warm Up Lesson Presentation Lesson Quiz

  2. 26 ANSWER 2.Find the midpoint ofCA. (0, 5) ANSWER Warm-Up In Exercises 1– 4, use A(0, 10),B(24, 0), and C(0, 0). 1.FindAB.

  3. 3.Find the midpoint of AB. (12, 5) ANSWER 4.Find the slope ofAB. 5 – ANSWER 12 Warm-Up In Exercises 1– 4, use A(0, 10),B(24, 0), and C(0, 0).

  4. CONSTRUCTION Triangles are used for strength in roof trusses. In the diagram, UVand VWare midsegments of RST. Find UV and RS. 1 1 2 2 RT ( 90 in.) = 45 in. = = UV VW ( 57 in.) 2 2 = 114 in. = = RS Example 1 SOLUTION

  5. ANSWER UW 2. In Example 1, suppose the distance UWis 81 inches. Find VS. 81 in. ANSWER Guided Practice 1. Copy the diagram in Example 1. Draw and name the third midsegment.

  6. 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 SOLUTION

  7. Example 3 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. b. Notice that you need to use three different variables. Example 3 a. Let hrepresent the length and krepresent the width.

  9. In Example 2, if Fis the midpoint of CB, what do you know about DF? ANSWER DF is a midsegment of ABC. DF ABandDFis half the length of AB. Guided Practice 3.

  10. 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 Guided Practice

  11. 5. Is it possible to find any of the side lengths in part (b) of Example 3 without using the Distance Formula? Explain. 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. (m, m) ANSWER Guided Practice

  12. 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 Place an isosceles right triangle in a coordinate plane. Then find the length of the hypotenuse and the coordinates of its midpoint M. SOLUTION

  13. = = = = 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 Use the Distance Formula to find PQ. PQ = Use the Midpoint Formula to find the midpoint Mof the hypotenuse.

  14. 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 Write a coordinate proof of the Midsegment Theorem for one midsegment.

  15. 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

  16. 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 Guided Practice 7.

  17. 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 Guided Practice 8.

  18. 26 10 ANSWER ANSWER Lesson Quiz Use the figure for Exercises 1–4. 1. If UV = 13, find RT. 2. If ST = 20, find UW.

  19. 34 in. 6 ANSWER ANSWER Lesson Quiz Use the figure for Exercises 1–4. 3. If the perimeter of RST = 68 inches, find the perimeter of UVW. 4. If VW = 2x – 4, and RS = 3x – 3, what isVW?

  20. ANSWER Lesson Quiz 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.

More Related