# Midpoint Formula

• MHB
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

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

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

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