Dividing Line Segments into Four Equal Parts using Midpoint Formula

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In summary: 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: $\sqrt{(2- 0)^2+ (2- 0)^2}= \sqrt{4+ 4}= \sqrt{4(2)}= 2\sqrt{2}$.
  • #1
nycmathdad
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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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  • #2
Beer soaked ramblings follow.
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
 
  • #3
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 $\frac{\sqrt{2}}{2}$.
 
  • #4
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?
 
  • #5
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: $\sqrt{(2- 0)^2+ (2- 0)^2}= \sqrt{4+ 4}= \sqrt{4(2)}= 2\sqrt{2}$.
 
  • #6
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.
 

Related to Dividing Line Segments into Four Equal Parts using Midpoint Formula

What is the Midpoint Formula?

The Midpoint Formula is a mathematical formula used to find the midpoint between two points on a coordinate plane. It is commonly used in geometry and can also be applied in other fields such as physics and engineering.

How is the Midpoint Formula calculated?

The Midpoint Formula is calculated by finding the average of the x-coordinates and the average of the y-coordinates of the two given points. The coordinates of the midpoint will be the averages of the x and y coordinates respectively.

Can the Midpoint Formula be used for any type of coordinates?

Yes, the Midpoint Formula can be used for any type of coordinates, as long as they are given in pairs of x and y coordinates. This includes both positive and negative coordinates, as well as decimals and fractions.

What is the significance of the Midpoint Formula?

The Midpoint Formula is significant because it allows us to find the center point between two given points. This can be useful in various applications, such as finding the center of a circle or determining the average position of an object that moves between two points.

Are there any limitations to using the Midpoint Formula?

One limitation of the Midpoint Formula is that it can only be used to find the midpoint between two points. It cannot be used for more than two points or for finding the midpoint along a curve or non-linear path. Additionally, it is important to ensure that the given points are in the same coordinate plane when using this formula.

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