MHB Dividing Line Segments into Four Equal Parts using Midpoint Formula

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To divide a line segment into four equal parts, the Midpoint Formula is applied three times to find the necessary points. For the segments defined by the points (1, -2) and (4, -1), as well as (-2, -3) and (0, 0), the midpoints are calculated to determine the division points. The discussion highlights the importance of understanding the distance between points, which is derived from the distance formula. The calculations demonstrate how to find the midpoints of each segment and subsequently the points that divide the segments into equal lengths. This method effectively allows for precise division of line segments in coordinate geometry.
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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)

Looking for hints to solve 55 and 56.
 
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nycmathdad said:
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)

Looking for hints to solve 55 and 56.
Interpolate Country Boy's explanation at
https://mathhelpboards.com/threads/midpoint-formula.28521/#post-124867
 
The three points needed are the midpoint, p, of the given interval and the midpoint of the two intervals having one of the original endpoint and p as endpoints and the other original endpoint and p as endpoints.

For example, if an interval has endpoints (0, 0) and (2, 2), of length $\sqrt{2}$, has midpoint (1, 1). The midpoint of the interval from (0, 0) to (1, 1) is (1/2, 1/2) and the mid point of (1, 1) to (2, 2) is (3/2, 3/2). The four intervals from (0, 0) to (1/2, 1/2), from (1/2, 1/2) to (1, 1), from (1, 1) To (3/2, 3/2), and from (3/2, 3/2) all have length $\frac{\sqrt{2}}{2}$.
 
Country Boy said:
The three points needed are the midpoint, p, of the given interval and the midpoint of the two intervals having one of the original endpoint and p as endpoints and the other original endpoint and p as endpoints.

For example, if an interval has endpoints (0, 0) and (2, 2), of length $\sqrt{2}$, has midpoint (1, 1). The midpoint of the interval from (0, 0) to (1, 1) is (1/2, 1/2) and the mid point of (1, 1) to (2, 2) is (3/2, 3/2). The four intervals from (0, 0) to (1/2, 1/2), from (1/2, 1/2) to (1, 1), from (1, 1) To (3/2, 3/2), and from (3/2, 3/2) all have length $\frac{\sqrt{2}}{2}$.

Interesting. By the length sqrt{2}, you mean the distance between two given points. This is found using the distance formula for points. True?
 
Yes, although I miswrote. I was first thinking of (0, 0) to (1, 1) which does have length $\sqrt{2}$. But then I changed to (0, 0) to (2, 2) which is twice as long: $\sqrt{(2- 0)^2+ (2- 0)^2}= \sqrt{4+ 4}= \sqrt{4(2)}= 2\sqrt{2}$.
 
Country Boy said:
Yes, although I miswrote. I was first thinking of (0, 0) to (1, 1) which does have length $\sqrt{2}$. But then I changed to (0, 0) to (2, 2) which is twice as long: $\sqrt{(2- 0)^2+ (2- 0)^2}= \sqrt{4+ 4}= \sqrt{4(2)}= 2\sqrt{2}$.

Ok. Interesting.
 
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