Chapter 11

Lesson

11-5

The Pythagorean Distance Formula

Vocabulary taxicab distance

BIG IDEA From the Pythagorean Theorem, a formula can be obtained for the distance between any two points in the plane.

In Lesson 11-4, you saw that by putting figures on the coordinate plane, you could prove that certain lines are parallel or perpendicular. This lesson shows how to determine whether segments on the coordinate plane are congruent.

Mental Math

Distances on the Coordinate Plane

a. 27¢.

State the fewest number of coins (pennies, nickels, dimes, quarters) you could have if you had b. $1.45.

In some rural areas of the United States, there are roads 1 mile apart going north, south, east, and west. Juan’s house is 3 miles east and 5 miles north of Cierra’s house, as shown on the grid at the right with Cierra’s house at point (0, 0) and Juan’s house at (3, 5).

c. 99¢.

When Cierra goes from her house to Juan’s house along the roads, she travels 8 miles. This distance, along horizontal and vertical lines, is sometimes called the taxicab distance from one place to another.

y

A bird flying from ___ Cierra’s house to Juan’s house would fly directly along CJ, taking the shortest distance. This distance is the length of the hypotenuse of a right triangle with legs 3 miles and 5 miles. It can be found using the Pythagorean Theorem. It is the distance “as the crow flies” and is the usual distance between points in geometry. You can also think of this distance as the magnitude of a vector starting at (0, 0) and ending at (3, 5).

3

J (3,5)

5 4

2 1

C (0,0)

1

2

3 (3,0)

4

x

QY1 Any time that you need to find the distance between two points on the coordinate plane on an oblique line, you can draw a right triangle whose hypotenuse is the distance between the two points.

GUIDED

Example 1 Find the distance d between (–26, 7) and (28, 25) to the nearest tenth.

676

QY1

Use the Pythagorean Theorem to find the distance between Cierra’s and Juan’s houses, to the nearest tenth of a mile.

Indirect Proofs and Coordinate Proofs

SMP_SEGEO_C11L05_676-681.indd 676

5/27/08 4:02:40 PM

Lesson 11-5 y

Solution Plot the points. Extend a vertical line from one point 30

and a horizontal line from the other and find the coordinates of the third vertex of a right triangle. The lengths of the legs are ? and ? .

20


(᎑26, 7)

10

Now, use the Pythagorean Theorem. d2 =

? ?

2


(28, 25)

x

+ ?

᎑30

2

᎑20

᎑10

10

20

30

= ? d = √ d≈ ? The above process can always be used to calculate distances on an oblique line, but it is very useful to have a general formula. Let P = (x1, y1) be the first point and ___ R = (x 2, y 2) be the second point. First find a point Q so that PR is the hypotenuse___ of a right triangle PQR. Such____ a point is Q = (x 2, y1). Then PQ is a horizontal segment and QR is a vertical segment. You can view these segments as on a horizontal or vertical number line and thus their lengths are ⎪x 2 - x1⎥ and ⎪y 2 - y1⎥ respectively. So,

y

P =
(x1, y1) |x2 - x1|

Q =
(x2, y1)

|y2 - y1|

PR 2 = PQ 2 + QR 2

x

= ⎪ x 2 - x 1⎥ 2 + ⎪ y 2 - y 1⎥ 2 = (x 2 - x1

)2

+ (y 2 - y1

R =
(x2, y2)

) 2.

Taking the square roots of both sides, PR =

(x 2 - x1)2 + ( y 2 - y1)2 . √

This is a formula you should learn. We call it the Pythagorean Distance Formula to distinguish it from taxicab distance. That name also reminds you of its origin. But most people just call it the “Distance Formula.” When we do not mention a type of distance between points A and B, and when we write AB, we mean this Pythagorean distance.

Theorem (Pythagorean Distance Formula on the Coordinate Plane) The distance d between two points (x1, y1) and (x2, y2) on the coordinate plane is (x2 - x1)2 + (y2 - y1)2 . d = √

With the Pythagorean Distance Formula, the distance of Guided Example 1 can be calculated without drawing a figure. QY2

QY2

Use the Distance Formula to find the distance in Example 1.

The Pythagorean Distance Formula

SMP_SEGEO_C11L05_676-681.indd 677

677

5/21/08 3:51:43 PM

Chapter 11

GUIDED

Example 2 Find the distance between (–5, 7) and (–15, 24), to the nearest tenth. Solution Substitute (–5, 7) and (–15, 24) into the distance formula.

d=

(x2 - x1)2 + (y2 - y1)2 √

= √ ( ? - ? )2 + ( ? - ? )2 Substitute for x1, x2, y1, and y2. ( ? )2 + ( ? )2 = √

Work first inside parentheses.

100 + 289 = √

Simplify.

389 = √

≈ 19.7 QY3

QY3

Using the Distance Formula in Proofs With the Distance Formula, you can prove that segments on the coordinate plane are congruent by showing their lengths are equal. Example 3 contains a coordinate proof of a theorem you have known for some time.

Use the Distance Formula to find the magnitude of the vector from Cierra’s house to Juan’s house at the beginning of this lesson.

Example 3 Using coordinates, prove that the diagonals of an isosceles trapezoid are congruent. Solution First state the given and what you need to prove in terms of

a figure. Given: TRAP is an isosceles trapezoid. Prove: AT = PR

One of the convenient locations for any isosceles trapezoid TRAP is with the y-axis as its symmetry line. Let T = (a, 0), P = (– a, 0), A = (–b, c), and R = (b, c).

y

A =
(᎑b, c)

R =
(b, c)

Now draw a figure. Using the Pythagorean Distance Formula, AT =

=

(–b - a)2 + (c - 0)2 √ (–b - a)(–b - a) + c2 √

= √ b2 + 2ab + a2 + c2

678

x

P =
(᎑a, 0)

T =
(a, 0)

Indirect Proofs and Coordinate Proofs

SMP_SEGEO_C11L05_676-681.indd 678

5/21/08 3:52:04 PM

Lesson 11-5 PR =

=

(b - – a)2 + (c - 0)2 √ (b + a)(b + a) + (c - 0)2 √

= √ b2 + 2ab + a2 + c2 Thus, AT = PR.

Notice that the coordinate proof is straightforward, but it does require knowledge of algebra and square roots.

Questions COVERING THE IDEAS y

ln 1−3, use the drawing at the right.

S =
(᎑5, 11)

1. a. What are the coordinates of Z? b. Find the taxicab distance between S and N. c. Find SN using the lengths ZN and ZS. 2. a. What are the coordinates of L? b. Find the taxicab distance between L and A. c. Find LA using the Pythagorean Distance Formula. d. Find KA. e. Find LK + KA. 3. Find SN using the Pythagorean Distance Formula.

10

A =
(2, 11) Y =
(3, 11)

K =
(᎑1, 8) 5

Z

N =
(3, 3)

L

x ᎑5

0

5

4. Give the distance between (a, b) and (c, d) in terms of a, b, c, and d. 5. Multiple Choice The Pythagorean Distance Formula gives you the distance between two points. A along horizontal and vertical segments. B as the crow flies. C as a taxicab would go. 6. BIG has B = (–600, 100), I = (1000, –100), and G = (300, 800). Is BIG scalene, isosceles but not equilateral, or equilateral? 7. To get to a hospital from the middle of a nearby town, you can drive 7 miles east, turn right and go 12 miles south, and then turn right again and go 2 miles west. By helicopter, how far is it from the middle of the town to the hospital? (Ignore the altitude of the helicopter.) 8. Use a coordinate proof to show that the diagonals of a rectangle are congruent.

The Pythagorean Distance Formula

SMP_SEGEO_C11L05_676-681.indd 679

679

5/21/08 3:52:21 PM

Chapter 11

APPLYING THE MATHEMATICS 9. On a map, it can be seen that Brandon lives 1 mile east and 0.2 mile south of school, while Mollie lives 0.4 mile west and 0.8 mile south of school. a. Draw a graph with appropriate coordinates for the school, Brandon’s residence, and Mollie’s residence. b. Using taxicab distance, who lives closer to the school, Brandon or Mollie? c. Using Pythagorean distance, who lives closer to the school? 10. An ant starts at a foot of a staircase and crawls to the base of the fourth step. If each step is 8 inches tall and 10 inches wide, how far is the ant from where it began? 11. Let A = (–2, 3) and B = (2, 4). Is the point C = (–1, 7) on, inside, or outside the circle with center A that contains B? 12. A football coach explains a “shoot route” passing play in which the receiver sprints up the field 4 yards, then turns –90º and runs for 2 yards, then turns 90º and runs up the field. If the receiver runs 7 yards after his second turn, how far did he run and how far is he from where he was when the play began?

10 yd 7 5 yd 2 line of scrimmage

4

13. In the video game below, the mouse runs 2 units east, 3 units north, 2 units east, 3 units north, 2 units east and then 5 units south. How far is the mouse from where it began?

14. Let A = (x1, y1) and B = (x 2, y 2). Let S be a size transformation of magnitude 5. Show that the distance between S(A) and S(B) is 5 times the distance between A and B. 15. C and D are points on the coordinate plane. The x-coordinate of D is 4 less than the x-coordinate of C. The y-coordinate of D is 2 more than the y-coordinate of C. Find CD. 16. a. Find the exact magnitude of a vector from (–4, 5) to (7, –11). b. Find the magnitude of a vector from (d, e) to (f, g).

680

Indirect Proofs and Coordinate Proofs

SMP_SEGEO_C11L05_676-681.indd 680

5/21/08 3:52:34 PM

Lesson 11-5

REVIEW 17. Let A = (3, 4), B = (1.2, –6), C = (–11, –3.2), and D = (–7, 9). Prove that ABCD is not a trapezoid. (Lesson 11-4) 18. Suppose that PENTA is a pentagon in the coordinate plane, and that none of its vertices are on the x- and y-axes. Use the Law of Ruling Out Possibilities to show that at least two of the vertices of PENTA lie in the same quadrant. (Lessons 11-2, 11-1) 19. a. What does it mean for two segments to bisect each other? (Lesson 2-4)

b. Name all of the special types of quadrilaterals whose diagonals bisect each other. (Lesson 7-9)

20. a. Describe how you could divide the interior of a regular n-gon into n congruent regions. b. Describe how you could divide the interior of a regular 12-gon into 4 congruent regions. c. Suppose n = k, where k and  are integers. Describe how you could divide the interior of a regular n-gon into k congruent regions. (Lesson 6-8) 21. Let A be the set of all points (x, y) in the coordinate plane such that x and y are both integers. What is the image of A under translation by the vector (–3, 2)? (Lesson 4-6) 22. What is the mean of x and y? (Previous Course) EXPLORATION 23. Let C = (10, 15) and A = (7, –2). Suppose E, I, O, and U are four other points with AC = EC = IC = OC = UC. Give possible coordinates of E, I, O, and U.

QY ANSWERS

1. d 2 = 32 + 52 = 34, so, d = √ 34 ≈ 5.8 mi 2. d = (28 - (–26))2 + (25 - 7)2 √ 542 + 182 = √ 3240 = √ ≈ 56.9 3. d = (3 - 0)2 + (5 - 0)2 = √ √ 32 + 52 = √ 34 ≈ 5.8 The Pythagorean Distance Formula

SMP_SEGEO_C11L05_676-681.indd 681

681

5/21/08 3:53:05 PM