Distance & midpoint formulas 11.5

P. 740 1-15, 35

16 pts.

P. 740 1-15, 35

1. Hypotenuse

2. If the sum of 9 squared and 12 squared is 15 squared, then it is a right triangle. If they are not equal, then it is not a right triangle.

3. b = 4

4. a = 6.3

5. c = 7.8

6. a = 8.77. c = 11.38. c = 139. c = 14.410. b = 2411. a = 812. b = 4013. a = 1.6

14. b = 3.5

15. C 85.9 feet

16 pts total

Distance & Midpoint Formulas

11.5

P. 744

• In this section, you will – learn how to find the distance between two

point in a coordinate plane

– Learn how to find the midpoint between two point in a coordinate plane

Both use formulas to calculate the answer.

• Distance Formula

• Midpoint Formula1 2 1 2( , )

2 2

x x y y

2 22 1 2 1( ) ( )d x x y y

Distance Formula

• Find the distance between (1,4) & (-2,3)

• D is about 3.16

2 22 1 2 1( ) ( )d x x y y

EXAMPLE 1 Find the distance between two points

Find the distance between (– 1, 3) and (5, 2)

d ( x2 – x1 )2 + ( y2 – y1)2= Distance formula

= (5 – (–1))2 + ( 2 – 3)2 Substitute.

= 62 + (– 1)2 = 37 Simplify.

ANSWER

The distance between the points is units.37

2 22 1 2 1( ) ( )d x x y y

Find the distance between (3, -1) and (4,0)

• Decide whether the point (3,2), (2,0) (-1,4) are vertices of a right triangle.

• d1

• d2

• d3

Midpoint Formula

• The midpoint between (x1,y1) and (x2,y2) is:

• Find the midpoint between (-2,3) & (4,2)

• Check with a graph• (1, 2 ½)

1 2 1 2( , )2 2

x x y y

Midpoint Formula

1 2 1 2( , )2 2

x x y y

Find the midpoint between

(5, -1) and (-3, 7)

Find the midpoint between

(-1,2) and (7,4)

EXAMPLE 3 Standardized Test Practice

SOLUTION

Let ( x1, y1 ) = ( –1, – 2) and ( x2, y2 ) = ( 3, – 4 ).

( )x1 + x2 y1 + y2

2 2, = ( )

– 1 + 3 – 2 + (– 4) 2 2

, Substitute

= (1,– 3) Simplify

ANSWER

The correct answer is B.

• Find the midpoint of (-4,1) and (2,-3)

• (-1,-1)

• Assignment: P. 747

3-10, 22-27

Make sure you put formulas - followed by work for each problem. Round to the tenths

• Semester Test will be ….Do some reviewing this next week! Also, we’ll be working

different word problems on RTD

P. 747 3-10, 22-27

3. 1 4. 35. 56. 5.47. 7.38. 8.19. 8.210. 16.3

22. (4,2)

23. (5,-5)

24. (-2,7)

25. (1,-4)

26. (-1,1)

27. (-11,-6) 14 pts

Determine if the points (-2,3) (-1,1) and (2,3) are vertices of a right triangle.

EXAMPLE 4 Solve a real-world problem

SIGHTSEEING

You and a friend are sightseeing in Washington, D.C. You are at the National Gallery of Art, and your friend is at the Washington Monument, as shown on the map. You want to meet at the landmark that is closest to the midpoint of your locations. At which landmark should you meet?

EXAMPLE 4 Solve a real-world problem

SOLUTION

Your coordinates are (11, 3), and your friend’s coordinates are (2, 2). First, find the midpoint of your locations, which is

x1 + x2 y1 + y2

2 2( ), ( ) 11 + 2 3 + 2

2 2

,= = (6.5, 2.5).

Next, find the distance from the midpoint to the Smithsonian Institution, located at (7, 1), and to the Natural History Museum, located at (7, 3).

EXAMPLE 4 Solve a real-world problem

ANSWER

You should meet at the Natural History Museum.

Distance to Smithsonian Institution:

d =

(6.5 – 7)2 + (2.5 – 1)2

1.58 units

Distance to Natural History Museum:

0.71 unit(6.5 – 7)2 + (2.5 – 3)2d =

• Worksheet 11.5

• 1-6, 16-21, 22-24, 28

• You may want to put work in your notebook.

Wksh 11.5 1-6,16-24,28

1. 2.2

2. 3.2

3. 7.1

4. 10

5. 6.1

6. 10.4

16. (3, 8.5)

17.(-8.5,8)

18. (-1, -9.5)

19. (5.5, -1.5)

20. (25, 0)

21.(-1.5, 2)

22.Yes

23.Yes

24.No

28.About 14-15 miles

16 points

Extra Cr

29. a.(1750,2000)

b. 1953 ft

30. 8.5 Books