Midpoint Formula...2?

Question

55. Use the Midpoint Formula three times to find the three points that divide the line segment joining (x_1, y_1) and (x_2, y_2) into four

equal parts.

56. Use the result of Exercise 55 to find the points that divide each line segment joining the given points into four equal parts.

(a) (x_1, y_1) = (1, −2)

(x_2, y_2) = (4, −1)

(b) (x_1, y_1) = (−2, −3)

(x_2, y_2) = (0, 0)

I need help with 55 and 56.

llaffer · Accepted Answer

Given your points:

(x₁, y₁) and (x₂, y₂)

The midpoint formula is to get the means of the x's and the means of the y's.

The first time you do this will get the middle point between the two endpoints:

[ (x₁ + x₂) / 2,  (y₁ + y₂) / 2 ]

that's the middle point.  Now we do the same thing two more times using the first point and the midpoint as the endpoints to find the middle point that is the 1/4 point.  Then again starting from the midpoint to the second point to be the 3/4 point.

1/4 point - x:

[x₁ + (x₁ + x₂) / 2] / 2

Simplify that, starting with a common denominator:

[2x₁ / 2 + (x₁ + x₂) / 2] / 2

Add the numerators:

[(2x₁ + x₁ + x₂) / 2] / 2
[(3x₁ + x₂) / 2] / 2

Turn the division of fractions into the multiplication of the reciprocal and simplify:

[(3x₁ + x₂) / 2] * 1 / 2
(3x₁ + x₂) / 4

That's the x of the 1/4 point.  Now for the y.  Same steps:

[y₁ + (y₁ + y₂) / 2 ] / 2
[2y₁ / 2 + (y₁ + y₂) / 2 ] / 2
[(2y₁ + y₁ + y₂) / 2 ] / 2
[(3y₁ + y₂) / 2 ] / 2
(3y₁ + y₂) / 4

So the 1/2 point is:

[ (3x₁ + x₂) / 4, (3y₁ + y₂) / 4 ]

You can do the rest from here to get the 3/4 point, then using the equations for your second part.

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Midpoint Formula...2?

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