Find the distance between (−4, −3) and (−2, 0). Round your answer to the nearest tenth.
1 answer:
Answer:Given the two points you entered of (-4, -3) and (-2, 0), we need to calculate 8 items:
Item 1: Calculate the slope and point-slope form:
Slope (m) = y2 - y1
x2 - x1
Slope (m) = 0 - -3
-2 - -4
Slope (m) = 3
2
Calculate the point-slope form using the formula below:
y - y1 = m(x - x1)
y - -3 = 3/2(x + 4)
Item 2: Calculate the line equation that both points lie on.
The standard equation of a line is y = mx + b where m is our slope, x and y are points on the line, and b is a constant.
Rearranging that equation to solve for b, we get b = y - mx. Using the first point that you entered = (-4, -3) and the slope (m) = 3/2 that we calculated, let's plug in those values and evaluate:
b = -3 - (3/2 * -4)
b = -3 - (-12/2)
b = -6
2
-
-12
2
b = 6
2
This fraction is not reduced. Using our GCF Calculator, we see that the top and bottom of the fraction can be reduced by 6
Our reduced fraction is:
1
0.33333333333333
Now that we have calculated (m) and (b), we have the items we need for our standard line equation:
y = 3/2x + 3
Item 3: Calculate the distance between the 2 points you entered.
Distance = Square Root((x2 - x1)2 + (y2 - y1)2)
Distance = Square Root((-2 - -4)2 + (0 - -3)2)
Distance = Square Root((22 + 32))
Distance = √(4 + 9)
Distance = √13
Distance = 3.6056
Item 4: Calculate the Midpoint between the 2 points you entered. Midpoint is denoted as follows:
Midpoint =
x2 + x1
2
,
y2 + y1
2
Midpoint =
-4 + -2
2
,
-3 + 0
2
Midpoint =
-6
2
,
-3
2
Midpoint = (-3, -3/2)
Item 5: Form a right triangle and calculate the 2 remaining angles using our 2 points:
Using our 2 points, we form a right triangle by plotting a 3rd point (-2,-3)
Our first triangle side = -2 - -4 = 2
Our second triangle side = 0 - -3 = 3
Using the slope we calculated, Tan(Angle1) = 1.5
Angle1 = Atan(1.5)
Angle1 = 56.3099°
Since we have a right triangle, we only have 90 degrees left, so Angle2 = 90 - 56.3099° = 33.6901
Item 6: Calculate the y intercept of our line equation
The y intercept is found by setting x = 0 in the line equation y = 3/2x + 3
y = 3/2(0) + 3
y = 3
Item 7: Determine the parametric equations for the line determined by (-4, -3) and (-2, 0)
Parametric equations are written in the form (x,y) = (x0,y0) + t(b,-a)
Plugging in our numbers, we get
(x,y) = (-4,-3) + t(-2 - -4,0 - -3)
(x,y) = (-4,-3) + t(2,3)
x = -4 + 2t
y = -3 + 3t
Calculate Symmetric Equations:
x - x0
z
y - y0
b
Plugging in our numbers, we get:
x - -4
2
y - -3
3
Step-by-step explanation:
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