Question

Find the distance between (−4, −3) and (−2, 0). Round your answer to the nearest tenth.

Answer 1

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: