August 29, 2016 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

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

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

1.3 Use Midpoint and Distance Formulas WARM - UP

Measure the distance between the endpoints and the midpoint. What do you notice?

Vocabulary midpoint – the point that divides the segment into two congruent segments segment bisector – a point, ray, line, line segment, or plane that intersects the segment at its midpoint

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 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. STEP 2 Evaluate the expression for VM when x = 4. VM = 4x – 1 = 4(4) – 1 = 15 So, the length of VM is 15.

EXAMPLE 2 Use algebra with segment lengths 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 In Exercises 1 and 2, identify the segment bisector of PQ . Then find PQ. 1. 3 4 ANSWER MN;

GUIDED PRACTICE for Examples 1 and 2 In Exercises 1 and 2, identify the segment bisector of PQ . Then find PQ. 2. line l ; 11 5 7 ANSWER

Graph Activity Plot (-2, 4) and (3, 4). Fold the two points and find the midpoint. Now plot (3, 3) and (3, -2). Now, find the midpoint between (1, -3) and (4, 2).

Graph Activity Find the midpoint between (1, -3) and (4, 2).

The Midpoint Formula The coordinates of the midpoint of a segment are the averages of the x- and y-coordinates of the endpoints If A(x1,y1) and B(x2,y2) are points in a coordinate plane, then the midpoint M of AM has coordinates

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. SOLUTION a. 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.

EXAMPLE 3 Use the Midpoint Formula SOLUTION 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. 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)

WARM - UP September 1, 2016 1. The endpoints of AB are A(3, -4) and B(-6, 5). Find the coordinates of the midpoint M. 2. The endpoints of AB are A(-4, -8) and B(7, 6). Find the coordinates of the midpoint M. 3. The midpoint of VW is M(-4, – 9). One endpoint is W(8, 4). Find the coordinates of endpoint V. 4. The midpoint of VW is M(5, 10). One endpoint is W(-12, 6). Find the coordinates of endpoint V.

Distance Formula To find the distance between the two points, use the distance formula

Distance Formula

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 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 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 B ANSWER

Daily Homework 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

Daily Homework 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)

Daily Homework 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

Class Work Sect. 1.3 #’s 3,4,10,13,18,23,24,25,33,34,36,42,43 46,47,49,53 Homework Sect. 1.3 #’s 6,8,9,14,20,26,35,45,49,52

More Examples What is the measure of segment PR if Q is the midpoint of segment PR?

Answer If Q is the midpoint then segments PQ and QR are both equal to 6 – 3x So, 2(6 – 3x) = 14x + 2 12 – 6x = 14x + 2 10 = 20x x = ½ PR = 14(1/2) + 2 = 7 + 2 = 9

Last Example What is the measure of segment AC if B is the midpoint of segment AC?

1.3 Use Midpoint

1.3 Use Midpoint

Distance Formula Write this into your foldable B (x2, y2) C(x2, y1) • A(x1, y1) |x2 – x1| |y2 – y1| Find AB (AB)2 = (x2 – x1)2 + (y2 – y1)2 This formula can be used to find the distance between any two points in the plane

1.3 Distance Formula

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