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)

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

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

EXAMPLE 2 Use the Midsegment Theorem 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.

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 ? ANSWER DF is a midsegment of ABC. DF AB and DF is 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

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

EXAMPLE 4 Apply variable coordinates 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).

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

EXAMPLE 5 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 SOLUTION STEP 1 Place OBC and assign coordinates. Because you are finding midpoints, use 2p, 2q, and 2r. Then find the coordinates of D and E. D( ) 2q + 0, 2r + 0 2 = D(q, r) E( ) 2q + 2p, 2r + 0 E(q+p, r)

EXAMPLE 5 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 . STEP 3 Prove DE = OC. Use the Ruler Postulate 1 2 to find DE and OC . DE = (q + p) – q = p OC = 2p – 0 = 2p So, the length of DE is half the length of OC

GUIDED PRACTICE for Examples 4 and 5 7. In Example 5, find the coordinates of F, the midpoint of OC . Then show that EF OB . (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 ANSWER

GUIDED PRACTICE for Examples 4 and 5 8. 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 yes; OJ = m, JH = n, HO = m2 + n2, OJ: ( , 0), JH: (m, ), HO: ( , ) 2 m n Sample:

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

Daily Homework Quiz 4. If VW = 2x – 4, and RS = 3x – 3, what is VW? ANSWER 6 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