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Pre-AP Geometry Worksheet 1.3

Pre-AP Geometry Name: ___________________ Worksheet 1.3: Midpoint and Distance Date: _______ Period: ____ For problems 1-4, you will have two answers. Show all work.

a) Find the other endpoint, point “B”, given endpoint “A” and midpoint “M” b) Find the distance between points “A” & “B”. Leave answers in simplified radical form.

1) A ( 2, -1) M (5, 5) 2) A (3, 2) M (-5, -2) 3) A (5, 4) M (1, -2) 4) A (-2, 3) M (3, -2) 5)Which point on the coordinate grid satisfies 6) If a crow leaves Marcus and flies 3 miles

the conditions of x ≥ 3 and y ≤ -1? due north, then turns and flies 4 miles due east, how far is he from Marcus?

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Pre-AP Geometry Worksheet 1.3

Use PEMDAS = Parentheses, Exponents, Multiplication/Division, Add/Subtract to simplify the following.

7) 1•5− 6 ÷ 2+ 32 8)

125÷ 5 2+ 3( )⎡⎣ ⎤⎦ 9) 4+ 2(10− 4•6) 10) 3(2+ 7)

2 ÷5 Use Algebraic reasoning to solve the following equations.

11)

12

x −10 = −13 12) 3x −8 = 2x + 9

13) 5x −8+ 2x = 6 14) −3(x − 7) = 2x + 4

PRACTICE and APPLICATION EXERCISES

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Find the length of each segment.

1. AB 2. BD

Use the figure at the right for Exercises 3 and 4.

3. If RS = 15 and ST = 9, then RT = ■. 4. If ST = 15 and RT = 40, then RS = ■. 5. Apply Mathematics (1)(A) The numbers labeled on the map of Florida are mile

markers. Assume that Route 10 between Quincy and Jacksonville is straight.

Suppose you drive at an average speed of 55 mi/h. How long will it take to get from Live Oak to Jacksonville?

6. On a number line, A is at - 2 and B is at 4. What is the coordinate of C, which is 23 of the way from A to B?

7. Analyze Mathematical Relationships (1)(F) A is the midpoint of XY .

a. Find XA.

b. Find AY and XY.

8. Suppose point E has a coordinate of 3 on <EG

> and EG = 5. What are the possible

coordinates of point G?

Use the diagram at the right for Exercises 9 and 10.

9. If AD = 12 and AC = 4y - 36, find the value of y. Then find AC and DC.

10. If ED = x + 4 and DB = 3x - 8, find ED, DB, and EB. 11. Explain Mathematical Ideas (1)(G) Suppose you know PQ and QR.

Can you use the Segment Addition Postulate to find PR? Explain.

12. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Use the diagram at the right.

a. What algebraic expression represents GK?

b. If GK = 30, what are GH and JK?

Determine the coordinate of the midpoint of the segment with the given endpoints.

13. 2 and 4 14. -9 and 6 15. 2 and -5

SR T

Quincy

181Tallahassee

MonticelloMadison

Live OakLake City

MacclennyJacksonville

199

225251 283 303

335357

10

10

95

95

X A Y

3x 5x ! 6

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14 Lesson 1-2 Measuring Segments

A B C D E

731!6!8

D

A

E C

B

x2x ! 3

4x " 3

G H J K

PRACTICE and APPLICATION EXERCISES

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HO M E W O

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Find the coordinates of the midpoint of HX .

1. H(0, 0), X(8, 4) 2. H(-1, 3), X(7, -1) 3. H(13, 8), X(-6, -6) 4. H(7, 10), X(5, -8) 5. H(-6.3, 5.2), X(1.8, -1) 6. H(5 12, -4 34 ), X (2 14, -1 14 ) 7. Explain Mathematical Ideas (1)(G) The endpoints of AB are A( - 2, - 3) and

B(3, 2). Point C lies on AB and is 25 of the way from A to B. What are the coordinates of point C? Explain how you found your answer.

8. Do you use the Midpoint Formula or the Distance Formula to find the following?

a. Given points K and P, find the distance from K to the midpoint of KP.

b. Given point K and the midpoint of KP, find KP.

For each graph, find (a) AB to the nearest tenth and (b) the coordinates of the midpoint of AB.

9. 10. 11.

12. Create Representations to Communicate Mathematical Ideas (1)(E) Graph the points A(2, 1), B(6, -1), C(8, 7), and D(4, 9). Draw parallelogram ABCD, and diagonals AC and BD.

a. Find the midpoints of AC and BD.

b. What appears to be true about the diagonals of a parallelogram?

Apply Mathematics (1)(A) The units of the subway map at the right are in miles. Suppose the routes between stations are straight. Find the distance you would travel between each pair of stations to the nearest tenth of a mile.

13. Oak Station and Jackson Station

14. Central Station and South Station

15. Elm Station and Symphony Station

16. Cedar Station and City Plaza Station

17. Maple Station is located 6 mi west and 2 mi north of City Plaza. What is the distance between Cedar Station and Maple Station?

6y

x6

A

B

!6!9 O

y

x4

A

B

4 8

!6

O!2 O

!2

1

3

B

Ay

x2

South

Central

North

City Plaza

Jackson

Cedar

Oak

Elm

x

y6

!6

4

!4

2

2 4!2 O!4

Symphony

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196 Lesson 5-1 Midpoint and Distance in the Coordinate Plane

Find the distance between each pair of points. If necessary, round to the nearest tenth.

18. J(2, -1), K(2, 5) 19. L(10, 14), M(-8, 14) 20. N(-1, -11), P(-1, -3) 21. A(0, 3), B(0, 12) 22. C(12, 6), D(-8, 18) 23. E(6, -2), F(-2, 4)

24. Explain Mathematical Ideas (1)(G) An airplane at T(80, 20) needs to fly to both U(20, 60) and V(110, 85). What is the shortest possible distance for the trip? Explain.

Use Representations to Communicate Mathematical Ideas (1)(E) For Exercises 25–29, use the map below. Find the distance between the cities to the nearest tenth.

25. Augusta and Brookline

26. Brookline and Charleston

27. Brookline and Davenport

28. Everett and Fairfield

29. List the cities in the order of least to greatest distance from Augusta.

The diagram shows AB with A(x1, y1), B(x2, y2), C(x1 + x22 , y2), and D(x1 + x22 , y1). 30. Are ∠C and ∠D right angles? Justify your answer.

31. Are △ADM and △BCM congruent? Explain.

32. Use triangle congruency to derive the formula for finding the coordinates of the midpoint M of AB.

33. Select Techniques to Solve Problems (1)(C) The endpoints of FH are F (–7, 7) and H (1, 3). Point G lies on FH between F and H, such that FG = 35 FH . Select a technique (such as estimation or number sense) that will help you calculate the coordinates of point G, and explain your selection. What are the coordinates of point G?

Use the diagram for Exercises 34 and 35. Points P(x1, y1) and Q(x2, y2) are the endpoints of a hypotenuse of a right triangle.

34. Derive a formula for the distance between P and Q.

35. A translation of PQ up n units results in ST with endpoints S(x1, y1 + n) and T (x2, y2 + n). Derive a formula for finding ST. Explain your reasoning. How are PQ and ST related?

36. The endpoints of AB are A (–3, –2) and B (9, 4). Points D and E lie on AB, between A and B. A classmate says that AD = 23AB and AE =

56AB.

What are the endpoints of DE?

0 4

4

8

12

8 12 16–4–8

y

x

Fairfield

Everett

Davenport

Charleston

BrooklineAugusta

AD

C B

M

O x1 x2

y1

y2

Q(x2, y2)

P (x1, y1)

R(x1, y2)y

O x

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