Definitions and Postulates

Apply the Segment Addition Postulate EXAMPLE 1 Apply the Segment Addition Postulate Measure the length of ST to the nearest tenth of a centimeter. SOLUTION Align one mark of a metric ruler with S. Then estimate the coordinate of T. For example, if you align S with 2, T appears to align with 5.4. ST = 5.4 – 2 = 3.4 Use Ruler Postulate. The length of ST is about 3.4 centimeters. ANSWER

EXAMPLE 2 Apply the the Segment Addition Postulate Maps The cities shown on the map lie approximately in a straight line. Use the given distances to find the distance from Lubbock, Texas, to St. Louis, Missouri. SOLUTION Because Tulsa, Oklahoma, lies between Lubbock and St. Louis, you can apply the Segment Addition Postulate. LS = LT + TS = 380 + 360 = 740 The distance from Lubbock to St. Louis is about 740 miles. ANSWER

Use a ruler to measure the length of the segment to the nearest inch. GUIDED PRACTICE for Examples 1 and 2 Use a ruler to measure the length of the segment to the nearest inch. 1 8 1. MN 1 in 5 8 ANSWER Use ruler postulate 2. PQ 1 in 3 8 ANSWER Use ruler postulate

In Exercises 3 and 4, use the diagram shown. GUIDED PRACTICE for Examples 1 and 2 In Exercises 3 and 4, use the diagram shown. 3. Use the Segment Addition Postulate to find XZ. SOLUTION xz = xy + yz Segment addition postulate = 23 + 50 Substitute 23 for xy and 50 for yz = 73 Add ANSWER xz = 73

GUIDED PRACTICE for Examples 1 and 2 In the diagram, WY = 30. Can you use the Segment Addition Postulate to find the distance between points W and Z? 4. NO; Because w is not between x and z. ANSWER

Use the diagram to find GH. EXAMPLE 3 Find a length Use the diagram to find GH. SOLUTION Use the Segment Addition Postulate to write an equation. Then solve the equation to find GH. FG + GH = FH Segment Addition Postulate. 21 + GH = 36 Substitute 36 for FH and 21 for FG. = 15 GH Subtract 21 from each side.

Compare segments for congruence EXAMPLE 4 Compare segments for congruence Plot J(– 3, 4), K(2, 4), L(1, 3), and M(1, – 2) in a coordinate plane. Then determine whether JK and LM are congruent. SOLUTION To find the length of a horizontal segment, find the absolute value of the difference of the x-coordinates of the endpoints. JK = 2 – (– 3) = 5 Use Ruler Postulate.

Compare segments for congruence EXAMPLE 4 Compare segments for congruence To find the length of a vertical segment, find the absolute value of the difference of the y-coordinates of the endpoints. LM = – 2 – 3 = 5 Use Ruler Postulate. JK and LM have the same length. So, JK LM. = ~ ANSWER

Use the diagram at the right to find WX. 5. GUIDED PRACTICE for Examples 3 and 4 Use the diagram at the right to find WX. 5. Use the segment addition postulate to write an equation. Then solve the equation to find WX SOLUTION vx = vw + wx Segment addition postulate 144 = 37 + wx Substitute 37 for vw and 144 for vx 107 = wx Subtract 37 from each side ANSWER WX = 107

GUIDED PRACTICE for Examples 3 and 4 6. Plot the points A(– 2, 4), B(3, 4), C(0, 2), and D(0, – 2) in a coordinate plane. Then determine whether AB and CD are congruent. Length of AB is not equal to the length of CD, so they are not congruent ANSWER

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 SOLUTION a. FIND MIDPOINT Use the Midpoint Formula. 2 5 1 + 4 – 3 + 2 = , M – 1 The coordinates of the midpoint M are 1 , – 5 2 ANSWER

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

( ) 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. SOLUTION Use the midpoint formula. ( ) 1 + 7 2 2 + 8 , M = M (4, 5) ANSWER The Coordinates of the midpoint M are (4,5).

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. SOLUTION Let (x, y) be the coordinates of endpoint V. Use the Midpoint Formula. STEP 1 Find x. STEP 2 Find y. 4+ x – 1 2 = 4+ y – 2 2 = 4 + x = – 2 4 + y = – 4 x = – 6 y = – 8 ANSWER The coordinates of endpoint V is (– 6, – 8)

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

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

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. 1 2 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

Use the Distance Formula. You may find it helpful to draw a diagram. 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 SOLUTION Use the Distance Formula. You may find it helpful to draw a diagram. (x – x ) + (y – y ) 2 1 AB = Distance Formula [2 –(–3)] + (–4 –1) 2 = Substitute. (5) + (5 ) 2 = Subtract.

GUIDED PRACTICE for Example 4 25+25 = 50 = 7.2 = Evaluate powers. 50 = Add. Use a calculator to approximate the square root. 7.2 = The correct answer is B ANSWER

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

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

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 1. In Exercises 1 and 2, identify the segment bisector of PQ . Then find PQ. SOLUTION 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. 7 8 1 = + Substitute 3 4 = Add.

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

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