Transcript Aim: How do we use the Pythagorean Theorem to find the

Aim: How do we use the Pythagorean
Theorem to find the distance between two
points?
Do Now:
In inches, the lengths of the
diagonals of a rhombus are 30
and 40. Find the length of one
side of the rhombus.
Aim: Distance Formula
Course: Applied Geometry
Pythagorean Theorem
What is the
measure of the
hypotenuse of
the right 
PRQ?
Q
5?
P
Q
5?
3
4
R
P
R
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 = 25
2 = 25, then c = 5.
3. If cAim:
Distance Formula
Course: Applied Geometry
Pythagorean Theorem
C
What is the
measure of the
hypotenuse of
right  ABC?
13
A
5
12
B
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 = 169
3. If c2 = 169, then c = 13.
Aim: Distance Formula
Course: Applied Geometry
Developing the Distance Formula
T
4
R
S
8
What is the length of RS?
8
What is the length of RT?
4
What is the length of ST?
1.
2.
3.
4.
c2 = a2 + b2 Pythagorean Theorem
ST2 = RS2 + RT2 Pythagorean Theorem
ST2 = 82 + 42 = 64 +16 = 80
2 = 80, then ST = 80
If ST
Aim: Distance Formula
Course: Applied Geometry
Developing the Distance Formula
B (4,6)
(1,1)A
3
5
C (4,1)
Plot the following coordinate points: A(1, 1),
B(4, 6) and connect them. How do we find
the length of AB?
1. AC = (4 - 1) = 3
BC = (6 - 1) = 5
2. AC = (xC - xA)
3. BC = (yB - yC)
4. (AB)2 = (AC)2 + (BC)2 Pythagorean Thr.
5. (AB)2 =
32 +
52 =
9 + 25 = 34
Aim:2Distance
Formula
6. If (AB)
= 34,
then AB =
34
Course: Applied Geometry
Developing the Distance Formula
(-2,1)A
45  3 5
3
(-2,-2) C
B (4,-2)
6
Plot the following coordinate points: A(-2,1),
B(4,-2) and connect them. What is the
length of AB?
1. Length of AC = (yA - yC) AC = (1 - -2) = 3
2. Length of BC = (xB - xC) BC = (4 - -2) = 6
Is distance a negative number?
NO!!
3. (AB)2 = (AC)2 + (BC)2 Pyth. Theorem
4. (AB)2 =
32 +
62
=
Aim:2 Distance
5. If (AB)
= 45,Formula
then AB =
9 + 36 = 45
45  3 5
Course: Applied Geometry
Developing the Distance Formula
(x2, y2)
b
(x2, y1)
d
a
(x1, y1)
(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
4. (d)2 = (x2 - x1)2 + (y2 - y1)2
2
2
5.AB
d  (x2  x1)  (y2  y1 )
Aim: Distance Formula
Course: Applied Geometry
Distance Formula
The Distance Formula
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.
d  ( x2  x1 )2  ( y2  y1 )2
(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).
d  ( 3  1) 2  ( 1  3) 2 
4 2  4 2  32
Aim: Distance Formula
Course: Applied Geometry
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)
2
2
d = AB  (x2  x1)  (y2  y1 )
2
2
AB  (2  1)  (7  2)
2
AB  (3)  (5)
AB 
9  25
AB 
34
Aim: Distance Formula
2
Course: Applied Geometry
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.
C(1,9)
B(8,9),
D(1,3)
A(8,3)
Show BD  AC
d  ( x 2  x1 ) 2  ( y 2  y1 ) 2
BD  (8 1) 2  (9  3)2
2
BD  7  6
AC  (7) 2  (6)2
2
BD  49  36
BD  85
AC  (8 1) 2  (3  9)2
AC  49  36
BD  AC AC  85
Aim: Distance Formula
Course: Applied Geometry
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.
d  ( x 2  x1 ) 2  ( y 2  y1 ) 2
2
2
AC  10  (k  0)  (0  0)
2
2
AC  10 
k 0
AC  10 
k2
AC  10 k
Aim: Distance Formula
Course: Applied Geometry
The Product Rule
Aim: Distance Formula
Course: Applied Geometry