Dividing Line Segments into Four Equal Parts using Midpoint Formula

[nycmathdad]

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.

 

[jonah1]

Beer soaked ramblings follow.

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

 

[HOI]

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 $\frac{\sqrt{2}}{2}$.

 

[nycmathdad]

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?

 

[HOI]

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: $\sqrt{(2- 0)^2+ (2- 0)^2}= \sqrt{4+ 4}= \sqrt{4(2)}= 2\sqrt{2}$.

 

[nycmathdad]

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.