Geometry Section 1-3 1112
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Transcript of Geometry Section 1-3 1112
Section 1-3Distance and Midpoints
Wednesday, September 28, 2011
Essential Questions
How do you find the distance between two points?
How do you find the midpoint of a segment?
Wednesday, September 28, 2011
Vocabulary
1. Pythagorean Theorem:
2. Distance:
Wednesday, September 28, 2011
Vocabulary
1. Pythagorean Theorem: a2 + b2 = c2 , where a and
b are legs of a right triangle, and c is the hypotenuse
2. Distance:
Wednesday, September 28, 2011
Vocabulary
1. Pythagorean Theorem: a2 + b2 = c2 , where a and
b are legs of a right triangle, and c is the hypotenuse
2. Distance: The length of the segment formed between two points
Wednesday, September 28, 2011
Vocabulary
1. Pythagorean Theorem: a2 + b2 = c2 , where a and
b are legs of a right triangle, and c is the hypotenuse
2. Distance: The length of the segment formed between two points
d = (x
2− x
1)2 + (y
2− y
1)2 for (x
1, y
1) and (x
2, y
2)
Wednesday, September 28, 2011
Vocabulary
3. Midpoint:
4. Segment Bisector:
Wednesday, September 28, 2011
Vocabulary
3. Midpoint: The point on a segment that is halfway between the endpoints
4. Segment Bisector:
Wednesday, September 28, 2011
Vocabulary
3. Midpoint: The point on a segment that is halfway between the endpoints
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟ for (x
1, y
1) and (x
2, y
2)
4. Segment Bisector:
Wednesday, September 28, 2011
Vocabulary
3. Midpoint: The point on a segment that is halfway between the endpoints
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟ for (x
1, y
1) and (x
2, y
2)
4. Segment Bisector: Any segment, line, or plane that intersects another segment at its midpoint
Wednesday, September 28, 2011
Example 1Use the number line to find DJ.
−4 −3−2 −1 0 1 2 3 4
J D
Wednesday, September 28, 2011
Example 1Use the number line to find DJ.
−4 −3−2 −1 0 1 2 3 4
J D
x
2− x
1
Wednesday, September 28, 2011
Example 1Use the number line to find DJ.
−4 −3−2 −1 0 1 2 3 4
J D
x
2− x
1 4 − (−4)
Wednesday, September 28, 2011
Example 1Use the number line to find DJ.
−4 −3−2 −1 0 1 2 3 4
J D
x
2− x
1 4 − (−4)
4 + 4
Wednesday, September 28, 2011
Example 1Use the number line to find DJ.
−4 −3−2 −1 0 1 2 3 4
J D
x
2− x
1 4 − (−4)
4 + 4
8
Wednesday, September 28, 2011
Example 1Use the number line to find DJ.
−4 −3−2 −1 0 1 2 3 4
J D
x
2− x
1 4 − (−4)
4 + 4
8 8
Wednesday, September 28, 2011
Example 1Use the number line to find DJ.
−4 −3−2 −1 0 1 2 3 4
J D
x
2− x
1 4 − (−4)
4 + 4
8 8
8 units
Wednesday, September 28, 2011
Example 2Graph A(3, 2) and B(6, 8). Then use the
Pythagorean Theorem to find AB.
Wednesday, September 28, 2011
Example 2Graph A(3, 2) and B(6, 8). Then use the
Pythagorean Theorem to find AB.
x
y
Wednesday, September 28, 2011
Example 2Graph A(3, 2) and B(6, 8). Then use the
Pythagorean Theorem to find AB.
x
y
A
Wednesday, September 28, 2011
Example 2Graph A(3, 2) and B(6, 8). Then use the
Pythagorean Theorem to find AB.
x
y
A
B
Wednesday, September 28, 2011
Example 2Graph A(3, 2) and B(6, 8). Then use the
Pythagorean Theorem to find AB.
x
y
A
B
Wednesday, September 28, 2011
Example 2Graph A(3, 2) and B(6, 8). Then use the
Pythagorean Theorem to find AB.
x
y
A
B
3
6
Wednesday, September 28, 2011
Example 2Graph A(3, 2) and B(6, 8). Then use the
Pythagorean Theorem to find AB.
x
y
A
B
3
6 a
2 + b2 = c2
Wednesday, September 28, 2011
Example 2Graph A(3, 2) and B(6, 8). Then use the
Pythagorean Theorem to find AB.
x
y
A
B
3
6 a
2 + b2 = c2
32 + 62 = c2
Wednesday, September 28, 2011
Example 2Graph A(3, 2) and B(6, 8). Then use the
Pythagorean Theorem to find AB.
x
y
A
B
3
6 a
2 + b2 = c2
32 + 62 = c2
9 + 36 = c2
Wednesday, September 28, 2011
Example 2Graph A(3, 2) and B(6, 8). Then use the
Pythagorean Theorem to find AB.
x
y
A
B
3
6 a
2 + b2 = c2
32 + 62 = c2
9 + 36 = c2
45 = c2
Wednesday, September 28, 2011
Example 2Graph A(3, 2) and B(6, 8). Then use the
Pythagorean Theorem to find AB.
x
y
A
B
3
6 a
2 + b2 = c2
32 + 62 = c2
9 + 36 = c2
45 = c2
45 = c2
Wednesday, September 28, 2011
Example 2Graph A(3, 2) and B(6, 8). Then use the
Pythagorean Theorem to find AB.
x
y
A
B
3
6 a
2 + b2 = c2
32 + 62 = c2
9 + 36 = c2
45 = c2
45 = c2
c ≈ 6.708203933 units
Wednesday, September 28, 2011
Example 3Use the distance formula to find the distance
between A(3, 2) and B(6, 8).
Wednesday, September 28, 2011
Example 3Use the distance formula to find the distance
between A(3, 2) and B(6, 8).
d = (x
2− x
1)2 + (y
2− y
1)2
Wednesday, September 28, 2011
Example 3Use the distance formula to find the distance
between A(3, 2) and B(6, 8).
d = (x
2− x
1)2 + (y
2− y
1)2
= (6 − 3)2 + (8 − 2)2
Wednesday, September 28, 2011
Example 3Use the distance formula to find the distance
between A(3, 2) and B(6, 8).
d = (x
2− x
1)2 + (y
2− y
1)2
= (6 − 3)2 + (8 − 2)2
= 32 + 62
Wednesday, September 28, 2011
Example 3Use the distance formula to find the distance
between A(3, 2) and B(6, 8).
d = (x
2− x
1)2 + (y
2− y
1)2
= (6 − 3)2 + (8 − 2)2
= 32 + 62
= 9 + 36
Wednesday, September 28, 2011
Example 3Use the distance formula to find the distance
between A(3, 2) and B(6, 8).
d = (x
2− x
1)2 + (y
2− y
1)2
= (6 − 3)2 + (8 − 2)2
= 32 + 62
= 9 + 36
= 45
Wednesday, September 28, 2011
Example 3Use the distance formula to find the distance
between A(3, 2) and B(6, 8).
d = (x
2− x
1)2 + (y
2− y
1)2
= (6 − 3)2 + (8 − 2)2
= 32 + 62
= 9 + 36
= 45
≈ 6.708203933 unitsWednesday, September 28, 2011
Example 4Find the midpoint of AB for points A(3, 2) and
B(6, 8).
Wednesday, September 28, 2011
Example 4Find the midpoint of AB for points A(3, 2) and
B(6, 8).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
Wednesday, September 28, 2011
Example 4Find the midpoint of AB for points A(3, 2) and
B(6, 8).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
=
3 + 62
,2 + 8
2
⎛
⎝⎜⎞
⎠⎟
Wednesday, September 28, 2011
Example 4Find the midpoint of AB for points A(3, 2) and
B(6, 8).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
=
3 + 62
,2 + 8
2
⎛
⎝⎜⎞
⎠⎟
=
92
,102
⎛
⎝⎜⎞
⎠⎟
Wednesday, September 28, 2011
Example 4Find the midpoint of AB for points A(3, 2) and
B(6, 8).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
=
3 + 62
,2 + 8
2
⎛
⎝⎜⎞
⎠⎟
=
92
,102
⎛
⎝⎜⎞
⎠⎟ =
92
,5⎛
⎝⎜⎞
⎠⎟
Wednesday, September 28, 2011
Example 4Find the midpoint of AB for points A(3, 2) and
B(6, 8).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
=
3 + 62
,2 + 8
2
⎛
⎝⎜⎞
⎠⎟
=
92
,102
⎛
⎝⎜⎞
⎠⎟ =
92
,5⎛
⎝⎜⎞
⎠⎟ or 4.5,5( )
Wednesday, September 28, 2011
Example 5Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
Wednesday, September 28, 2011
Example 5Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
Wednesday, September 28, 2011
Example 5Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
(−2,3) =
x + 82
,y + 6
2
⎛
⎝⎜⎞
⎠⎟
Wednesday, September 28, 2011
Example 5Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
(−2,3) =
x + 82
,y + 6
2
⎛
⎝⎜⎞
⎠⎟
−2 =
x + 82
Wednesday, September 28, 2011
Example 5Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
(−2,3) =
x + 82
,y + 6
2
⎛
⎝⎜⎞
⎠⎟
−2 =
x + 82
i2
Wednesday, September 28, 2011
Example 5Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
(−2,3) =
x + 82
,y + 6
2
⎛
⎝⎜⎞
⎠⎟
−2 =
x + 82
i2 (2)i( )
Wednesday, September 28, 2011
Example 5Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
(−2,3) =
x + 82
,y + 6
2
⎛
⎝⎜⎞
⎠⎟
−2 =
x + 82
i2 (2)i( )
−4 = x + 8
Wednesday, September 28, 2011
Example 5Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
(−2,3) =
x + 82
,y + 6
2
⎛
⎝⎜⎞
⎠⎟
−2 =
x + 82
i2 (2)i( )
−4 = x + 8
x = −12
Wednesday, September 28, 2011
Example 5Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
(−2,3) =
x + 82
,y + 6
2
⎛
⎝⎜⎞
⎠⎟
−2 =
x + 82
i2 (2)i( )
−4 = x + 8
x = −12
3 =
y + 62
Wednesday, September 28, 2011
Example 5Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
(−2,3) =
x + 82
,y + 6
2
⎛
⎝⎜⎞
⎠⎟
−2 =
x + 82
i2 (2)i( )
−4 = x + 8
x = −12
3 =
y + 62
i2
Wednesday, September 28, 2011
Example 5Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
(−2,3) =
x + 82
,y + 6
2
⎛
⎝⎜⎞
⎠⎟
−2 =
x + 82
i2 (2)i( )
−4 = x + 8
x = −12
3 =
y + 62
i2 2i
Wednesday, September 28, 2011
Example 5Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
(−2,3) =
x + 82
,y + 6
2
⎛
⎝⎜⎞
⎠⎟
−2 =
x + 82
i2 (2)i( )
−4 = x + 8
x = −12
3 =
y + 62
i2 2i
6 = y + 6
Wednesday, September 28, 2011
Example 5Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
(−2,3) =
x + 82
,y + 6
2
⎛
⎝⎜⎞
⎠⎟
−2 =
x + 82
i2 (2)i( )
−4 = x + 8
x = −12
3 =
y + 62
i2 2i
6 = y + 6
y = 0
Wednesday, September 28, 2011
Example 5Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).
M =x
1+ x
2
2,y
1+ y
2
2
⎛
⎝⎜
⎞
⎠⎟
(−2,3) =
x + 82
,y + 6
2
⎛
⎝⎜⎞
⎠⎟
−2 =
x + 82
i2 (2)i( )
−4 = x + 8
x = −12
3 =
y + 62
i2 2i
6 = y + 6
y = 0 U(−12,0)
Wednesday, September 28, 2011
Example 6Find PQ if Q is the midpoint of PR.
2x + 3 4x − 1P Q R
Wednesday, September 28, 2011
Example 6Find PQ if Q is the midpoint of PR.
2x + 3 4x − 1P Q R
2x + 3 = 4x − 1
Wednesday, September 28, 2011
Example 6Find PQ if Q is the midpoint of PR.
2x + 3 4x − 1P Q R
2x + 3 = 4x − 1
4 = 2x
Wednesday, September 28, 2011
Example 6Find PQ if Q is the midpoint of PR.
2x + 3 4x − 1P Q R
2x + 3 = 4x − 1
4 = 2x
x = 2
Wednesday, September 28, 2011
Example 6Find PQ if Q is the midpoint of PR.
2x + 3 4x − 1P Q R
2x + 3 = 4x − 1
4 = 2x
x = 2
PQ = 2x + 3
Wednesday, September 28, 2011
Example 6Find PQ if Q is the midpoint of PR.
2x + 3 4x − 1P Q R
2x + 3 = 4x − 1
4 = 2x
x = 2
PQ = 2x + 3
PQ = 2(2) + 3
Wednesday, September 28, 2011
Example 6Find PQ if Q is the midpoint of PR.
2x + 3 4x − 1P Q R
2x + 3 = 4x − 1
4 = 2x
x = 2
PQ = 2x + 3
PQ = 2(2) + 3
PQ = 4 + 3
Wednesday, September 28, 2011
Example 6Find PQ if Q is the midpoint of PR.
2x + 3 4x − 1P Q R
2x + 3 = 4x − 1
4 = 2x
x = 2
PQ = 2x + 3
PQ = 2(2) + 3
PQ = 4 + 3
PQ = 7 units
Wednesday, September 28, 2011
Check for Understanding
Take a look at p. 30 #1-12 to see if you know what you would need to do to solve the
problems
Wednesday, September 28, 2011
Problem Set
Wednesday, September 28, 2011
Problem Set
p. 31 #13-55 odd, 68, 69
“Learn from yesterday, live for today, hope for tomorrow. The important thing is not to stop
questioning.” - Albert EinsteinWednesday, September 28, 2011