Question

between the origin (0,0) and (4,-6) find the distance simplified radical and the midpoint

Answer 1

Simplified radical was suppose to Be in parantheses

Answer 2

distance formula is
\[\sqrt{(x2-x1)^{2}+(y2-y1)^{2}}\]

Answer 3

this explains how to solve the problem

Answer 4

To find the distance, given the two points P1 and P2 where P1 = (x1,y1) and P2 = (x2,y2), you use the distance formula:

d = √{(x1 - x2)² + (y1 - y2)²}

(Note: I am using symbol √ to indicate the square root)

To find the coordinates of the midpoint, you must:

1. Compute the slope (m) of the line that includes the two points where:

m = (y1 - y2)/(x1 - x2)

2. Using the slope and one of the points (I will use the first point but either will work), find the y-intercept (b) of the line:

b = y1 - m(x1)

You now have values for m and b, therefore, you write the equation of the line that runs through the two points P1 and P2 as:

Y = mX + b

You know that this line, also, contains the midpoint (P3) where P3 = (x3,y3)

This means that:

y3 = mx3 + b

Therefore:

P3 = (x3, mx3 + b)

Let us take a moment and review the values that we know. We know the values for, d, m, b, x2, y2, x1, and y1. We don't know the values for x3 and y3 but we have an equation that expresses y3 in terms of x3.

Returning to the distance formula but using half of the distance:

d/2 = √{(x1 - x3)² + (y1 - y3)²}

substituting in the expression for y3:

d/2 = √{(x1 - x3)² + (y1 - mx3 - b)²}

squaring both sides:

d²/4 = (x1 - x3)² + (y1 - mx3 - b)²

regrouping the last term:

d²/4 = (x1 - x3)² + ({y1 - b} - mx3)²


rewriting the square terms as products:

d²/4 = (x1 - x3)(x1 - x3) + ({y1 - b} - mx3)({y1 - b} - mx3)

Using the F.O.I.L method on the terms:

d²/4 = (x1² - 2x1[x3] + [x3]²) + ({y1 - b}² -2m{y1 - b}[x3] + m²[x3]²)

I will leave the solution of the above equation to you. When you have computed the actual values, it is a much simpler equation. You will probably use the quadratic formula to solve the resulting quadratic and you will get the two values for x3. Make sure that you use only the value that is between x1 and x2; the other value is an "aliased" root caused by squaring.

Now that you know the value of x3, substitute that value into the equation:

Y = mX + b

to obtain the value of y3.

You now have the coordinates of the midpoint, P3