1. Finding Missing Endpoints

2. Review: Finding the midpoint given two endpoints
Find the coordinates of the midpoint of a segment with endpoints R(-12, 8) and S(6, 12)
Midpoint Formula: 𝑥 2 + 𝑥 1 2 , 𝑦 2 + 𝑦 1 2
6+(−12) 2
= −6 2
12+8 2
= 20 2
=−3
=10
−3, 10

3. Now we are GIVEN the midpoint and ONE endpoint
Now we are GIVEN the midpoint and ONE endpoint. We are looking for the MISSING ENDPOINT!
Find the coordinates of the missing endpoint if P is the midpoint of NQ. P is at (6, 3) and N is at (5, 4).
Q(?, ?)
N(5, 4)
P(6, 3)
Midpoint Formula: 𝑥 2 + 𝑥 1 2 , 𝑦 2 + 𝑦 1 2
Normally, when we plug in the x values of our two endpoints, this piece of the midpoint formula gives us the x coordinate of the midpoint.
And this piece normally gives us the y coordinate of the midpoint when we plug in the y values of our two endpoints.
But, if we plug in our two points, P and N, into the formula like normal, we will be finding the midpoint between P and N, not the missing endpoint!

4. Midpoint Formula: 𝑥 2 + 𝑥 1 2 , 𝑦 2 + 𝑦 1 2
Find the coordinates of the missing endpoint if P is the midpoint of NQ. P is at (6, 3) and N is at (5, 4).
Q(?, ?)
N(5, 4)
P(6, 3)
Midpoint Formula: 𝑥 2 + 𝑥 1 2 , 𝑦 2 + 𝑦 1 2
Since this piece normally GIVES us the x coordinate of the midpoint, we should set it EQUAL to the x-coordinate of the midpoint (P) that we already have! Plug in the x-coordinate of the one endpoint (N) we have into x2.
Do the same for y. Set this piece of the midpoint formula EQUAL to the y-coordinate of the midpoint (P), and plug in the y value of the one endpoint (N) we have into y2.
Now we have: 𝑥 1 2 = and 𝑦 1 2 =3

5. The missing endpoint, Q, has coordinates (7, 2)
Now we have: 𝑥 1 2 = and 𝑦 1 2 =3
4+𝑦 1 2 =3
(2)
(2)
(2)
(2)
5+𝑥 1 2 =6
5 + x1 = 12
4 + y1 = 6 -5 -5 -4 -4
x1 = 7
y1 = 2
The missing endpoint, Q, has coordinates (7, 2)

6. The missing endpoint, Q, has coordinates (12, 23)
Find the coordinates of the missing endpoint if P is the midpoint of NQ. P is at (2, 12) and N is at (-8, 1).
Midpoint Formula: 𝑥 2 + 𝑥 1 2 , 𝑦 2 + 𝑦 1 2
1+𝑦 1 2 =12
−8+𝑥 1 2 =2
(2)
(2)
(2)
(2)
-8 + x1 = 4
1 + y1 = 24 +8 +8 -1 -1
x1 = 12
y1 = 23
The missing endpoint, Q, has coordinates (12, 23)

7. The missing endpoint, N, has coordinates (-10, 16)
Find the coordinates of the missing endpoint if P is the midpoint of NQ. P is at (-4, 7) and Q is at (2, -2).
Midpoint Formula: 𝑥 2 + 𝑥 1 2 , 𝑦 2 + 𝑦 1 2
−2+𝑦 1 2 =7
2+𝑥 1 2 =−4
(2)
(2)
(2)
(2)
2 + x1 = -8
-2 + y1 = 14 -2 -2 +2 +2
x1 = -10
y1 = 16
The missing endpoint, N, has coordinates (-10, 16)

8. The missing endpoint, N, has coordinates (-11, 10)
Find the coordinates of the missing endpoint if P is the midpoint of NQ. P is at (-3, 4) and Q is at (5, -2).
The missing endpoint, N, has coordinates (-11, 10)

9. Fractional Distance

10. To find a fractional distance, multiply the distance times the fractional distance.
Ex: Jenny is taking her dog on a 5 mile walk. She wants to stop to take a break after they have completed 2/3 of their walk. Where will she take her break?
5(2/3)
= 3.3 miles

11. To find the coordinates of a point a fractional distance from one point to another:
1. Find the distance between the x values of the two given points. (You can use the distance formula 𝑥 2 − 𝑥 1 )
2. Multiply this distance by the fractional distance.
3. Add the resulting number to the x value of the point you are going from. This is the x value of the point a fractional distance from the first point.
4. Repeat for the y values.

12. The coordinates of R are (3, -2)
Find the coordinates of point R that is 1/8 of the way from point S (2, -4) to point T (10, 12)
Distance between the x values:
Distance between the y values:
10−2
= 8 =8
12−(−4)
= 12+4
= 16
=16
Multiply the distance by the fractional distance:
Multiply the distance by the fractional distance:
8 1 8 =1 =2
Add to the x value of S:
Add to the y value of S:
2 + 1
= 3
-4 + 2
= -2
The coordinates of R are (3, -2)

13. The coordinates of R are (-6.5, 4.5)
Find the coordinates of point R that is ¾ of the way from point S (-8, -6) to point T (-10, -20)
−10−(−8)
= −10+8
= −2 =2
−20−(−6)
= −20+6
= −14
=14
2 3 4
=1.5
=10.5
= -6.5
= 4.5
The coordinates of R are (-6.5, 4.5)

14. The coordinates of R are (1.25, 4)
Find the coordinates of point R that is ¾ of the way from point S (-7, 1) to point T (4, 5)
The coordinates of R are (1.25, 4)