In the skateboard design, VW bisects XY at point T, and XT = 39.9 cm. Find XY. Skateboard SOLUTION EXAMPLE 1 Find segment lengths Point T is the midpoint of XY. So, XT = TY = 39.9 cm. XY = XT + TY = = 79.8 cm Segment Addition Postulate Substitute. Add.

SOLUTION EXAMPLE 2 Use algebra with segment lengths STEP 1 Write and solve an equation. Use the fact that that VM = MW. VM = MW 4x – 1 = 3x + 3 x – 1 = 3 x = 4 Write equation. Substitute. Subtract 3x from each side. Add 1 to each side. Point M is the midpoint of VW. Find the length of VM. ALGEBRA

EXAMPLE 2 Use algebra with segment lengths STEP 2 Evaluate the expression for VM when x = 4. VM = 4x – 1 = 4(4) – 1 = 15 So, the length of VM is 15. Check: Because VM = MW, the length of MW should be 15. If you evaluate the expression for MW, you should find that MW = 15. MW = 3x + 3 = 3(4) +3 = 15

GUIDED PRACTICE for Examples 1 and 2 M is midpoint and line MN bisects the line PQ at M. So MN is the segment bisector of PQ. So PM = MQ =1 7 8 PQ = PM + MQ = = Segment addition postulate. Substitute Add. In Exercises 1 and 2, identify the segment bisector of PQ. Then find PQ. 1. SOLUTION

GUIDED PRACTICE for Examples 1 and 2 In Exercises 1 and 2, identify the segment bisector of PQ. Then find PQ. 2. SOLUTION M is midpoint and line l bisects the line PQ of M. So l is the segment bisector of PQ. So PM = MQ

GUIDED PRACTICE for Examples 1 and 2 STEP 2 Evaluate the expression for PQ when x = 18 7 PQ = 5x – – 2x = 3x + 4 PQ = = Substitute for x Simplify. STEP 1 Write and solve an equation PM = MQ 5x – 7 = 11 – 2x 7x = 18 Write equation. Substitute. Add 2x and 7 each side. x = 18 7 Divide each side by 7.

EXAMPLE 3 Use the Midpoint Formula a. FIND MIDPOINT The endpoints of RS are R(1,–3) and S(4, 2). Find the coordinates of the midpoint M.

EXAMPLE 3 Use the Midpoint Formula – =, M, – 1 M The coordinates of the midpoint M are 1, – ANSWER SOLUTION a. FIND MIDPOINT Use the Midpoint Formula.

EXAMPLE 3 Use the Midpoint Formula FIND ENDPOINT Let (x, y) be the coordinates of endpoint K. Use the Midpoint Formula. STEP 1 Find x. 1+ x 2 2 = 1 + x = 4 x = 3 STEP 2 Find y. 4+ y 1 2 = 4 + y = 2 y = – 2 The coordinates of endpoint K are (3, – 2). ANSWER b. FIND ENDPOINT The midpoint of JK is M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K.

GUIDED PRACTICE for Example 3 3. The endpoints of AB are A(1, 2) and B(7, 8). Find the coordinates of the midpoint M. Use the midpoint formula. ( ) , M = M (4, 5) ANSWER The Coordinates of the midpoint M are (4,5). SOLUTION

GUIDED PRACTICE for Example 3 4. The midpoint of VW is M(– 1, – 2). One endpoint is W(4, 4). Find the coordinates of endpoint V. Let (x, y) be the coordinates of endpoint V. Use the Midpoint Formula. STEP 1 Find x. 4+ x – 1 2 = 4 + x = – 2 x = – 6 STEP 2 Find y. 4+ y – 2 2 = 4 + y = – 4 y = – 8 The coordinates of endpoint V is (– 6, – 8) SOLUTION

EXAMPLE 4 Standardized Test Practice Use the Distance Formula. You may find it helpful to draw a diagram.

EXAMPLE 4 Standardized Test Practice Distance Formula Substitute. Subtract. Evaluate powers. Add. Use a calculator to approximate the square root. (x – x ) + (y – y ) RS= [(4 – 2)] + [(–1) –3] 22 = (2) + (–4 ) 22 = 4+16 = 20 = 4.47 = The correct answer is C.

GUIDED PRACTICE for Example 4 5. In Example 4, does it matter which ordered pair you choose to substitute for (x, y ) and which ordered pair you choose to substitute for (x, y ) ? Explain No, when squaring the difference in the coordinate you get the same answer as long as you choose the x and y value from the some period ANSWER

GUIDED PRACTICE for Example 4 6. What is the approximate length of AB, with endpoints A(–3, 2) and B(1, –4)? 6.1 units 7.2 units 8.5 units 10.0 units Distance Formula Substitute. Subtract. (x – x ) + (y – y ) ABAB = [2 –(–3)] + (–4 –1) 2 2 = (5) + (5 ) 22 = Use the Distance Formula. You may find it helpful to draw a diagram. SOLUTION

GUIDED PRACTICE for Example 4 Evaluate powers. Add. Use a calculator to approximate the square root = 50 = 7.2 = The correct answer is BANSWER