aim: distance formula course: applied geometry do now: aim: how do we use the pythagorean theorem to...

Aim: Distance Formula Course: Applied Geometry

Do Now:

Aim: How do we use the Pythagorean Theorem to find the distance between two points?

In inches, the lengths of the diagonals of a rhombus are 30 and 40. Find the length of one side of the rhombus.


Pythagorean Theorem

P R

Q

?

Q

?

P R 4

3

What is the measure of the hypotenuse of the right PRQ?

Find the measure of the hypotenuse of a right triangle with legs of 3 and 4.1. c2 = a2 + b2 Pythagorean Theorem

2. c2 = 32 + 42

= 9 + 16 = 253. If c2 = 25, then c = 5.

5 5


Pythagorean Theorem

Find the measure of the hypotenuse of a right triangle with legs of 5 and 12.

1. c2 = a2 + b2 Pythagorean Theorem

2. c2 = 52 + 122 = 25 + 144 = 1693. If c2 = 169, then c = 13.

What is the measure of the hypotenuse of right ABC?

A B

C

12

5

13


Developing the Distance Formula

What is the length of RS?

What is the length of RT?

What is the length of ST?

R

T

S

8

4

1. c2 = a2 + b2 Pythagorean Theorem

3. ST2 = 82 + 42 = 64 +16 = 804. If ST2 = 80, then ST =

2. ST2 = RS2 + RT2 Pythagorean Theorem

80

8

4



Plot the following coordinate points: A(1, 1), B(4, 6) and connect them. How do we find the length of AB?

5. (AB)2 = 32 + 52 = 9 + 25 = 34

6. If (AB)2 = 34, then AB =

4. (AB)2 = (AC)2 + (BC)2 Pythagorean Thr.

34

2. AC = (xC - xA)

3. BC = (yB - yC)

1. AC = (4 - 1) = 3

A(1,1) C (4,1)

(4,6)B

3

5

BC = (6 - 1) = 5


A(-2,1)

B (4,-2)C(-2,-2)

45 3 53

6


Plot the following coordinate points: A(-2,1), B(4,-2) and connect them. What is the length of AB?

4. (AB)2 = 32 + 62 = 9 + 36 = 45

5. If (AB)2 = 45, then AB =

1. Length of AC = (yA - yC) 2. Length of BC = (xB - xC)

AC = (1 - -2) = 3

BC = (4 - -2) = 6

Is distance a negative number? NO!!

3. (AB)2 = (AC)2 + (BC)2 Pyth. Theorem

45 3 5



(x2, y2) , (x1, y1) are endpoints of a line segment. What is the length d of this line segment?1. Length of a = (x2 - x1)

2. Length of b = (y2 - y1)

3. (d)2 = (a)2 + (b)2 Pythagorean Theorem

(x2, y2)

(x1, y1)(x2, y1)

4. (d)2 = (x2 - x1)2 + (y2 - y1)2

d

ab

Distance Formula

AB (x2 x1)2 (y2 y1 )2

5. d


The Distance Formula

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

(x2, y2) are the coordinates of one point and (x1, y1) are the coordinates of the second point.

Ex: Find the distance between the points (3,-1) and (-2, 3).

3244

)31()13(

22

22

d

To find the distance between two points on the coordinate plane we use the distance formula. The distance formula is based on the Pythagorean Theorem.


Model Problem

Use the distance formula to find the distance between points A(-2, 7) and B(1, 2).

A(-2,7) B(1,2)

(x2, y2) (x1, y1)

AB ( 2 1)2 (7 2)2

AB (x2 x1)2 (y2 y1 )2

d =

AB ( 3)2 (5)2

AB 9 25

AB 34


Model Problem

Given rectangle ABCD with vertices A(8, 3), B(8, 9), and C(1, 9), and D(1, 3). Show the diagonals of the rectangle are congruent.

A(8,3)

B(8,9),C(1,9)

D(1,3)Show BD AC

BD (8 1)2 (9 3)2 AC (8 1)2 (3 9)2

AC (7)2 ( 6)2

AC 49 36

AC 85

BD 72 62

BD 85

BD 49 36

BD AC

212

212 )()( yyxxd


Model Problems

Which point lies furthest from the origin?

1) (0,-9) 2) (-7,6) 3) (8,5) 4) (-2,9)

The coordinates of the opposite vertices of quadrilateral ABCD are A(0, 0) and C(k, 0). If AC = 10, find the positive values of k.

212

212 )()( yyxxd

AC 10 (k 0)2 (0 0)2

AC 10 k2 02

AC 10 k2

AC 10 k


The Product Rule

aim: distance formula course: applied geometry do now: aim: how do we use the pythagorean theorem to...

Documents