Activity: The Rectangular Coordinate System
Determine the distance and the midpoint.
a. A (-4,4) and B (5₁-1)
b. C (-1₁-6) and D (-6,5)
C. E (2,4) and F (1₁-3)
d. G (-4,4) and H (-2,2)
2. Find the other endpoint.
a endpoint (-1,9) midpoint (-9, -11)
b. endpoint (2,5) midpoint (6,1)
c. endpoint (8,-10) midpoint (4,8)
d. endpoint (5,2) midpoint (1,6)
3. A scalene right triangle has vertices at (2₁-3) and (-2,-3)
and the 3rd vertex on the X-axis. Find the 3rd vertex.
Determine the distance and the midpoint.
a. A (-4,4) and B (5₁-1)
b. C (-1₁-6) and D (-6,5)
C. E (2,4) and F (1₁-3)
d. G (-4,4) and H (-2,2)
2. Find the other endpoint.
a endpoint (-1,9) midpoint (-9, -11)
b. endpoint (2,5) midpoint (6,1)
c. endpoint (8,-10) midpoint (4,8)
d. endpoint (5,2) midpoint (1,6)
3. A scalene right triangle has vertices at (2₁-3) and (-2,-3)
and the 3rd vertex on the X-axis. Find the 3rd vertex.
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Answer:
To determine the distance and midpoint between two points in the rectangular coordinate system, you can use the distance formula and midpoint formula, respectively.
1a. A (-4,4) and B (5₁-1):
Distance:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
d = √[(5 - (-4))^2 + (-1 - 4)^2]
d = √[9^2 + (-5)^2]
d = √[81 + 25]
d = √106
Midpoint:
M = ((x1 + x2)/2, (y1 + y2)/2)
M = ((-4 + 5)/2, (4 + (-1))/2)
M = (1/2, 3/2)
1b. C (-1₁-6) and D (-6,5):
Distance:
d = √[(-6 - (-1))^2 + (5 - (-6))^2]
d = √[(-5)^2 + (11)^2]
d = √[25 + 121]
d = √146
Midpoint:
M = ((-1 + (-6))/2, (-6 + 5)/2)
M = (-7/2, -1/2)
1c. E (2,4) and F (1₁-3):
Distance:
d = √[(1 - 2)^2 + (-3 - 4)^2]
d = √[(-1)^2 + (-7)^2]
d = √[1 + 49]
d = √50
Midpoint:
M = ((2 + 1)/2, (4 + (-3))/2)
M = (3/2, 1/2)
1d. G (-4,4) and H (-2,2):
Distance:
d = √[(-2 - (-4))^2 + (2 - 4)^2]
d = √[2^2 + (-2)^2]
d = √[4 + 4]
d = √8
Midpoint:
M = ((-4 + (-2))/2, (4 + 2)/2)
M = (-3, 3)
2a. endpoint (-1,9) midpoint (-9, -11):
To find the other endpoint, we can use the midpoint formula and solve for the missing coordinates:
(-1 + x)/2 = -9
(9 + y)/2 = -11
Solving these equations, we find:
x = -17
y = -31
The other endpoint is (-17, -31).
2b. endpoint (2,5) midpoint (6,1):
Using the midpoint formula:
(2 + x)/2 = 6
(5 + y)/2 = 1
Solving these equations, we find:
x = 10
y = -3
The other endpoint is (10, -3).
2c. endpoint (8,-10) midpoint (4,8):
Using the midpoint formula:
(8 + x)/2 = 4
(-10 + y)/2 = 8
Solving these equations, we find:
x = 0
y = 26
The other endpoint is (0, 26).
2d. endpoint (5,2) midpoint (1,6):
Using the midpoint formula:
(5 + x)/2 = 1
(2 + y)/2 = 6
Solving these equations, we find:
x = -3
y = 10
The other endpoint is (-3, 10).
3. A scalene right triangle has vertices at (2₁-3) and (-2,-3) and the 3rd vertex on the X-axis. Find the 3rd vertex:
Since the triangle is right-angled, the third vertex will have the same y-coordinate as one of the given vertices and an x-coordinate of 0 (on the X-axis).
The y-coordinate of the given vertices is -3. Therefore, the third vertex is (0, -3).
Step-by-step explanation:
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