The Distance Formula

Find the Missing Side (Answer Key)Use a, b, and c to label the legs and the hypotenuse of the triangle below.

Pythagorean Theorem

(Ieg)2 + (Ieg)2 = (hvpotenusev'

Find the missing side lengths. Round your answers to the nearest tenth.

6

x

x=8

k=12

7

9

z = -/130 ~ 11.4

m = -/48.25 ~ 6.9

5

7

y = -fi4 ~ 8.6

n

n = -/32 = 4J2 ~ 5.7

The DistanceFormula

Segment Length (Answer Key)

,,4~~...•

" ,.. ~1 ,.

;-;'"

I ...,...,

J ~ "&;

-- 0- ~ ~- -p - D 4 - - I.• 1P...,~"..., t.. J~

~ ,.;;:-v

K",~ L

j~'v

Completethe table below

SegmentChange in Change in

Segment Endpoint 1 Endpoint 2 x-values y-values What do you notice?Length IX2 - xli IY2 - Y11AS is the same as the

AB 5 units (4, 7) (4, 2) 0 5 difference betweenthe y-values.

EF is the same as theEF 11 units (-8, 3) (-8,-8) 0 11 difference between

the y-values.

IJ is the same as theIJ 6 units (-3, 2) (-3, -4) 0 6 difference between

the y-values.

CD is the same as theCD 7 units (1, -2) (8, -2) 7 0 difference between

the x-values.

GH is the same as theGH 6 units (-5,8) (1,8) 6 0 difference between

the x-values.

KL is the same as theKL 3 units (-1, -7) (2, -7) 3 0 difference between

the x-values.

The Distance Formula

Where Did That Come From? Part I (Answer Key)

J:,.,4..,n

::::;,...

~v.""

~L-

A- •..- 0- 9- B- -I:) - )- ~- 3- 1)_ ~ Ib ~ 10-A

,.,-~-v

r-

.~

::;n-;::

-'",.,"', Ir

x

Complete the table for your assigned triangle. Sample answers:

Triangle 1 Triangle 2 Triangle 3

• Plot the vertices • Plot the vertices • Plot the verticesA (-2,-1) and 8(4,7) A(7,-2)and 8(3,-5) A(-9,-3)and 8(3,-8)

• Use a straightedge to • Use a straightedge to • Use a straightedge to-

Creati ng the draw segment AS. draw segment AS. draw segment AS.

triangle • Create a right triangle • Create a right triangle • Create a right triangle-

with AS as its with AS as its with AS as itshypotenuse. hypotenuse. hypotenuse.

• Label the vertex of the • Label the vertex of the • Label the vertex of theright angle C. right angle C. right angle C.

What is the length8 4 12

of BC?What is the length

6 3 5of AC?

Apply the 82 + 62 = c2 42 + 32 = c2 122 + 52 = c'Pythagorean 100 = c' 25 = c' 169 = c'Theorem to

determine the ,,1100= R" 55 =R" ~169 = JC2length of AB. 10 = c 5=c 13 = c

1. Share the information about your triangle with your group members.2. Is it possible to use the coordinate plane to determine the length of A8?

No, it is not possible because the segment is diagonal to the axes.3. What do you notice about the hypotenuse of each right triangle and the distance from point

A to point 8 for each triangle? They are the same.

The Distance Formula

Where Did That Come From? Part II (Answer Key)

1. Use the triangles from Where Did That Come From? Part I and the sample given to completethe table below.

Endpoint Endpoint Length of Leg Length of Leg Length of the HypotenuseTriangle Using the Pythagorean

One Two One Two Theorem

1-5-31 = 8 1-7- 81= 15(-5-3)2 +(-7-8)2 =c2

Sample (3, 8) (-5, -7) ~(-5 - 3)2 + (-7 - 8)2 = cunits units

17 = c

14--21 = 6 17--11 = 8(4--2)2 +(7--1)2 = c2

1 (-2,-1) (4, 7) ~(4 - _2)2 + (7 - _1)2 = cunits units

10 = c

13-71= 4 1-5--21 = 3(3 - 7)2 + (-5 - _2)2 = c2

2 (7, -2) (3, -5) ~(3 - 7)2 + (-5 - _2)2 = cunits units

5=c

13--91 = 12 1-8--31 = 5(3 - -9Y + (-8 - _3)2 = c2

3 (-9, -3) (3, -8) ~(3 - _9)2 + (-8 - _3)2 = cunits units

13=c

Look at the graph below:

y

~----...F

4---~-------------------------+X

2. Sketch a right triangle with DE as the hypotenuse.

3. Label the vertex of the right angle F.

4. Label the coordinates of point F on the graph. (Xl' Y2)

The Distance Formula

5. How can you determine the length of OF?Subtract the v-coordinates and take the absolute value. IY2 - yll

6. How can you determine the length of EF?Subtract the x-coordinates and take the absolute value. IX2 - xli

7. Use the Pythagorean Theorem to find the length of DE.

(X2 - Xl)2 + (Y2 - yl)2 = c2

~(X2 - Xl)2 + (Y2 - Yl)2 = C

8. What does the equation you found in question 7 represent?This equation represents the distance from point D to point E.

9. Refer to the table in question 1 and the answers to questions 5-7 to complete the table below.

Endpoint Endpoint Length of Leg Length of Leg Length of the HypotenuseTriangle Using the PythagoreanOne Two One Two Theorem

1-5- 31= 8(-5 - 3)2 + (-7 - 8)2 = c2

Sample (3, 8) (-5,-7) 1-7 - 81= 15 units ~(-5-3)2 +(-7-8? =cunits17 = c

(X2 - Xl)2 + (Y2 - yl)2 = c'(x., v.) (X2, Y2) IX2 - Xli IY2 - Yll ~(X2 - Xl)2 + (Y2 - Yl)2 = C

The Distance Formula

Round Robin: Applying the Distance Formula (Answer Key)

1. Pass your paper to the person seated at your right and solve the first problem. You may workwith your group to solve the problem.

2. Upon completion of problem 1, pass the papers to the right again and solve the second problem.You may continue to work together to solve the problem.

3. Continue this process for problems 3 and 4.

1. Find the length of the diameter of circle Cwith the center at (-3, 2) and the point(1, 5) on the given circle.

}'

..,

~~

1/,,- ......•.

"/ ~ '\ pr-

I v ;I'~!-- -

I ~ '..•,

./ v

~c

\ J- 0 -~ - ,-p -o ~ - 3 - 'I

x-

" .., l/'"" ~ L...",oo ~

v

r-~

d = ~(-3 _1)2 + (2 _ 5)2

The radius is 5 units. The diameter isequal to 2 times the radius, so it is equalto 10 units.

Name:

2. Given the points A(1.5, 3.5), B(l, 2), andC(5, 1.5), verify that an isosceles triangleis formed.

y~

6

5

4 A

3r-~

V,r-,

2B =".

1C

x4 5

AC = ~(1.5 - 5)2+ (3.5 _1.5)2

AC = "'16.25AC ~ 4.03

BC = ~(1_5)2 +(2-1.5lBC = "'16.25BC ~ 4.03

Since AB = Be, the triangle formed isisosceles.

Name:

The Distance Formula

3. Verify that the midsegment EF oftrapezoid ABCD is equal to the average of

- -the lengths of AB and CD.

vI'

A

c,

AS = ~(_2_2)2 +(5-2tAS=fiS

AS=5

CD = ~(-6 - 2t +(3 -(-3>tCD = .)100

CD=10

The average of the lengths of AS and

CD: 5+10 = 7.52

EF = ~(-4 - 2)2+(4 - (_0.5»2

EF = .)56.25EF = 7.5

Since the values are equal, the average- -

of the lengths of AS and CD is equal to

the length of midsegment EF.

6,

5 L A-x[""IIi

"""--~4 ....•.

~ V~ ./3 i>r-,1./ r--....2 V ,

I..•••

F G-

. . 1 1 1 14. Given the POints F(12",14), L(12",42")'

1 1 1 1 .A(8-,4-), and G(8-,l-), verify that4 2 4 4

the diagonals of the rectangle FLAG arecongruent.y

~~~--~-.--.--.---.--.--.--,

234 5 6 7 8 9 x

(1 1)2 (1 1)2FA=~ 12"-84 + 14-42

FA = .)56.125FA f'Z 7.49

(1 1)2 (1 1)2LG= 12"-84 + 42-14

LG = .)56.125LG f'Z 7.49

Since FA = LG, the diagonals arecongruent.