Use Midpoint and Distance Formulas Warm Up Lesson Presentation Lesson Quiz

Warm-Up 1. Find a point between A(–3, 5) and B(7, 5). ANSWER Sample: (2, 5) 2. Find the average of –11 and 5. ANSWER –3

Warm-Up 3. Solve = 5. 2 x + 7 ANSWER 3 4. Find √30 to the nearest hundredth. ANSWER 5.48

Warm-Up 5. Find √5 + √20 to the nearest hundredth. ANSWER 6.71

Point T is the midpoint of XY . So, XT = TY = 39.9 cm. Example 1 Skateboard In the skateboard design, VW bisects XY at point T, and XT = 39.9 cm. Find XY. 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.

Point M is the midpoint of VW . Find the length of VM . ALGEBRA Example 2 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 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 Point M is the midpoint of VW . Find the length of VM . ALGEBRA SOLUTION 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 In Exercises 1 and 2, identify the segment bisector of PQ . Then find PQ. 1. 3 4 ANSWER MN;

Guided Practice In Exercises 1 and 2, identify the segment bisector of PQ . Then find PQ. 2. line ; 11 5 7 ANSWER

Example 3 a. FIND MIDPOINT The endpoints of RS are R(1,–3) and S(4, 2). Find the coordinates of the midpoint M. 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 b. FIND ENDPOINT The midpoint of JK is M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K. SOLUTION b. 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 3. The endpoints of AB are A(1, 2) and B(7, 8). Find the coordinates of the midpoint M. ANSWER (4,5) 4. The midpoint of VW is M(– 1, – 2). One endpoint is W(4, 4). Find the coordinates of endpoint V. ANSWER (– 6, – 8)

Example 4 SOLUTION Use the Distance Formula. You may find it helpful to draw a diagram.

Example 4 SOLUTION (x – x ) + (y – y ) RS = [(4 – 2)] + [(–1) –3] = 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 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 differences in the coordinates, you get the same answer as long as you choose the x and y values from the same point. SAMPLE ANSWER

Guided Practice 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 B ANSWER

Lesson Quiz 1. AB bisects CD at E. If CE = in., Find CD. 1 4 2 1 2 4 ANSWER in. 2. Point M is the midpoint of XY. Find XM. ANSWER 17

Lesson Quiz 3. Point M is the midpoint of PQ with endpoints P(2, – 6 ) and Q(– 8, 0). Find the coordinates of M. ANSWER (–3, –3) 4. The midpoint of GH is M(4, –1). One endpoint is G(5, 3) . Find the coordinates of H. ANSWER (3, –5)

Lesson Quiz 5. To find the distance between the swing and the sandbox in his backyard, Darren made a graph and found the coordinates of the swing to be (7, 2) and the coordinates of the sandbox to be (– 3, 8). Find the distance between the swing and the sandbox to the nearest tenth of a unit. 11.7 ANSWER