Obj. 5 Midpoint and Distance Formulas
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Transcript of Obj. 5 Midpoint and Distance Formulas
Obj. 5 Midpoint and Distance
The student will be able to (I can):
• Find the midpoint of two given points.
• Find the coordinates of an endpoint given one endpoint and a midpoint.and a midpoint.
• Find the distance between two points.
The coordinates of a midpoint are the averages of the coordinates of the endpoints of the segment.
C A T
1 3 21
2 2
− += =
-2 2 4 6 8 10
2
4
6
8
10
x
y
• (5, 6)D
O
G
-2
x-coordinate:
y-coordinate:
2 8 105
2 2
+= =
4 8 126
2 2
+= =
midpoint formula
The midpoint M of with endpoints A(x1, y1) and B(x2, y2) is found by
AB
1 12 2M , 2 2
yxx y+ +
A
B
y
y2
●
Maverage of y1 and y2
0
A
x1 x2
y1
average of x1 and x2
Example Find the midpoint of QR for Q(—3, 6) and R(7, —4)
x1 y1 x2 y2
Q(—3, 6) R(7, —4)
21x 3x 7 42
2 2 2
+ += = =−
21 21
yy 6 4+ +=
−= =21 2
1y
2 2
y 6
2
4+ +=
−= =
M(2, 1)
Problems 1. What is the midpoint of the segment joining (8, 3) and (2, 7)?
A. (10, 10)
B. (5, —2)
C. (5, 5)
D. (4, 1.5)
8 2 105
2 2
+= =
3 7 105
2 2
+= =
Problems 2. What is the midpoint of the segment joining (—4, 2) and (6, —8)?
A. (—5, 5)
B. (1, —3)
C. (2, —6)
D. (—1, 3)
4 6 21
2 2
− += =
Problem 3. Point M(7, —1) is the midpoint of , where A is at (14, 4). Find the coordinates of point B.
A. (7, 2)
B. (—14, —4)
C. (0, —6)
D. (10.5, 1.5)
AB
D. (10.5, 1.5)
14 7 7− = 7 7 0− =
( )4 1 5− − = 1 5 6− − = −
Pythagorean Theorem
In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
2 2 2 22 2or b c b(ca a )+ = = +y
x
a
bc
●
●a
22 2c ba= +22c ba= +
22 164 93= + = +25 5= =
●
distance formula
Given two points (x1, y1) and (x2, y2), the distance between them is given by
Example: Use the Distance Formula to find the distance between F(3, 2) and G(-3, -1)
( ) ( )2
1
2
2 2 1d xx y y= − + −
x1 y1 x2 y2
3 2 —3 —13 2 —3 —1
( ) ( )2 2FG 3 3 1 2= − − + − −
( ) ( )2 26 3 36 9= − + − = +
45 6.7= ≈Note: Remember that the square of a negative number is positivepositivepositivepositive.
Problems 1. Find the distance between (9, —1) and (6, 3).
A. 5
B. 25
C. 7
D. 13
( ) ( )( )22d 6 9 3 1= − + − −
( )2 23 4 25 5= − + = =
Problems 2. Point R is at (10, 15) and point S is at (6, 20). What is the distance RS?
A. 1
B.
C. 41
D. 6.5
41
( ) ( )2 2d 6 10 20 15= − + −
( )2 24 5 41= − + =