Every triangle has ____ midsegments. The midsegments of ABC at the Section 5.1 Midsegment Theorem and Coordinate Proof A _______________ of a triangle is a ____________ that ____________ the ______________ of two sides of a triangle. Every triangle has ____ midsegments. The midsegments of ABC at the right are _______, _______, and _______. midsegment segment connects midpoints 3 MP MN NP

A midsegment connects midpoints Section 5.1 Midsegment Theorem and Coordinate Proof Theorem Theorem 5.1: Midsegment Theorem A midsegment connects midpoints of two sides and is parallel to the 3rd side. A midsegment length is half the 3rd side.

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

GUIDED PRACTICE for Example 1 1. Refer to the diagram in Example 1. Name the third midsegment. UW 2. In Example 1, suppose the length UW is 81 inches. Find VS. ST = 2(81) ST = 162, VS = (162) VS = 81 in.

EXAMPLE 2 Use the Midsegment Theorem In the kaleidoscope image, AE BE and AD CD . Show that CB DE . SOLUTION Both E and D are midpoints of AB and AC. DE is a midsegment of ABC by definition and CB DE by the Midsegment Theorem.

Section 5.1 Midsegment Theorem and Coordinate Proof COORDINATE PROOF A coordinate proof involves placing geometric figures in a _______________. When you use ____________ to represent the ______________ of a _______ in a coordinate plane, the ________ are ______ for all figures of that type. coordinate plane variables coordinates figure results true

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. a. A rectangle b. A scalene triangle 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 a. Let h represent the length and k represent the width. b. Notice that you need to use three different variables.

GUIDED PRACTICE for Examples 2 and 3 3. In Example 2, if F is the midpoint of CB , what do you know about DF ? DF is a midsegment of ABC. DF AB and DF is half the length of AB. 4. A square has vertices (0, 0), (m, 0), and (0, m). Find the fourth vertex. (m, m) (0, m) (0, 0) (m, 0)

EXAMPLE 4 Prove the Midsegment Theorem Write a coordinate proof of the Midsegment Theorem for one midsegment. GIVEN : DE is a midsegment of OBC. PROVE : DE OC and DE = OC 1 2 STEP 1: Find coordinates of D and E. Because you are finding midpoints, use 2p, 2q, and 2r. D( ) 2q + 0, 2r + 0 2 E( ) 2q + 2p, 2r + 0 2 = D(q, r) = E(q+p, r)

EXAMPLE 4 Prove the Midsegment Theorem STEP 2 Prove DE OC . The y-coordinates of D and E are the same, so DE has a slope of 0. OC is on the x-axis, so its slope is 0. Because their slopes are the same, DE OC . 1 2 STEP 3 Prove DE = OC. Find the lengths of DE and OC . DE = (q + p) – q = p OC = 2p – 0 = 2p So, the length of DE is half the length of OC

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