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Midpoint worksheet

Grades 9–10 · CCSS HSG.GPE.B.6

Solve each problem and show your work. Round answers to 2 decimal places where needed.

1.
Problem 1.
- Find: A missing endpoint B, from A and the midpoint
- Coordinates: 2D (x, y)
- Point A: x: 3
- Point A: y: 3
- Midpoint M: x: 4
- Midpoint M: y: 6
Find: Missing endpoint B
Answer
2.
Problem 2.
- Find: A missing endpoint B, from A and the midpoint
- Coordinates: 3D (x, y, z)
- Point A: x: 1
- Point A: y: 3
- Point A: z: 2
- Midpoint M: x: 6
- Midpoint M: y: 6
- Midpoint M: z: 2
Find: Missing endpoint B
Answer
3.
Problem 3.
- Find: A missing endpoint B, from A and the midpoint
- Coordinates: 2D (x, y)
- Point A: x: 2
- Point A: y: 4
- Midpoint M: x: 6
- Midpoint M: y: 10
Find: Missing endpoint B
Answer
4.
Problem 4.
- Find: The midpoint of A and B
- Coordinates: 2D (x, y)
- Point A: x: 3
- Point A: y: 3
- Point B: x: 10
- Point B: y: 9
Find: Midpoint M
Answer
5.
Problem 5.
- Find: A missing endpoint B, from A and the midpoint
- Coordinates: 3D (x, y, z)
- Point A: x: 2
- Point A: y: 2
- Point A: z: 1
- Midpoint M: x: 3
- Midpoint M: y: 7
- Midpoint M: z: 3
Find: Missing endpoint B
Answer
6.
Problem 6.
- Find: The midpoint of A and B
- Coordinates: 3D (x, y, z)
- Point A: x: 2
- Point A: y: 3
- Point A: z: 1
- Point B: x: 9
- Point B: y: 10
- Point B: z: 6
Find: Midpoint M
Answer
7.
Problem 7.
- Find: A missing endpoint B, from A and the midpoint
- Coordinates: 3D (x, y, z)
- Point A: x: 2
- Point A: y: 4
- Point A: z: 1
- Midpoint M: x: 4
- Midpoint M: y: 8
- Midpoint M: z: 3
Find: Missing endpoint B
Answer
8.
Problem 8.
- Find: The point dividing AB in a ratio m : n
- Coordinates: 3D (x, y, z)
- Point A: x: 2
- Point A: y: 2
- Point A: z: 1
- Point B: x: 8
- Point B: y: 9
- Point B: z: 4
- Ratio part m (from A): 1
- Ratio part n (to B): 2
Find: Point P dividing AB in the ratio 1 : 2
Answer
9.
Problem 9.
- Find: A missing endpoint B, from A and the midpoint
- Coordinates: 2D (x, y)
- Point A: x: 2
- Point A: y: 3
- Midpoint M: x: 5
- Midpoint M: y: 6
Find: Missing endpoint B
Answer
10.
Problem 10.
- Find: A missing endpoint B, from A and the midpoint
- Coordinates: 3D (x, y, z)
- Point A: x: 2
- Point A: y: 3
- Point A: z: 1
- Midpoint M: x: 6
- Midpoint M: y: 8
- Midpoint M: z: 3
Find: Missing endpoint B
Answer

Answer key: Midpoint worksheet

1. Missing endpoint B: (5, 9)
1. M is halfway, so the missing end B sits as far past M as A sits before it. Double each M number and subtract A's
  $B = (2xM − xA, 2yM − yA)$
2. Find the x-coordinate of B: twice M's, minus A's
  $xB$ $= 2 \times 4 - 3$ $= 5$
3. Find the y-coordinate of B: twice M's, minus A's
  $yB$ $= 2 \times 6 - 3$ $= 9$
4. Put the coordinates together to get the point B
  $B = (5, 9)$
5. For reference, find the length of AB: square each change, add them and take the square root
  $AB$ $= \sqrt{{(5 - 3)}^{2} + {(9 - 3)}^{2}}$ $= 6.324555$
2. Missing endpoint B: (11, 9, 2)
1. M is halfway, so the missing end B sits as far past M as A sits before it. Double each M number and subtract A's
  $B = (2xM − xA, 2yM − yA, 2zM − zA)$
2. Find the x-coordinate of B: twice M's, minus A's
  $xB$ $= 2 \times 6 - 1$ $= 11$
3. Find the y-coordinate of B: twice M's, minus A's
  $yB$ $= 2 \times 6 - 3$ $= 9$
4. Find the z-coordinate of B: twice M's, minus A's
  $zB$ $= 2 \times 2 - 2$ $= 2$
5. Put the coordinates together to get the point B
  $B = (11, 9, 2)$
6. For reference, find the length of AB: square each change, add them and take the square root
  $AB$ $= \sqrt{{(11 - 1)}^{2} + {(9 - 3)}^{2} + {(2 - 2)}^{2}}$ $= 11.661904$
3. Missing endpoint B: (10, 16)
1. M is halfway, so the missing end B sits as far past M as A sits before it. Double each M number and subtract A's
  $B = (2xM − xA, 2yM − yA)$
2. Find the x-coordinate of B: twice M's, minus A's
  $xB$ $= 2 \times 6 - 2$ $= 10$
3. Find the y-coordinate of B: twice M's, minus A's
  $yB$ $= 2 \times 10 - 4$ $= 16$
4. Put the coordinates together to get the point B
  $B = (10, 16)$
5. For reference, find the length of AB: square each change, add them and take the square root
  $AB$ $= \sqrt{{(10 - 2)}^{2} + {(16 - 4)}^{2}}$ $= 14.422205$
4. Midpoint M: (6.5, 6)
1. The midpoint M is halfway, so average the matching coordinates of the two ends A and B
  $M = ((xA + xB)/2, (yA + yB)/2)$
2. Add the two x-coordinates and divide by 2
  $xM$ $= \frac{3 + 10}{2}$ $= 6.5$
3. Add the two y-coordinates and divide by 2
  $yM$ $= \frac{3 + 9}{2}$ $= 6$
4. Put the coordinates together to get the point M
  $M = (6.5, 6)$
5. For reference, find the length of AB: square each change, add them and take the square root
  $AB$ $= \sqrt{{(10 - 3)}^{2} + {(9 - 3)}^{2}}$ $= 9.219544$
5. Missing endpoint B: (4, 12, 5)
1. M is halfway, so the missing end B sits as far past M as A sits before it. Double each M number and subtract A's
  $B = (2xM − xA, 2yM − yA, 2zM − zA)$
2. Find the x-coordinate of B: twice M's, minus A's
  $xB$ $= 2 \times 3 - 2$ $= 4$
3. Find the y-coordinate of B: twice M's, minus A's
  $yB$ $= 2 \times 7 - 2$ $= 12$
4. Find the z-coordinate of B: twice M's, minus A's
  $zB$ $= 2 \times 3 - 1$ $= 5$
5. Put the coordinates together to get the point B
  $B = (4, 12, 5)$
6. For reference, find the length of AB: square each change, add them and take the square root
  $AB$ $= \sqrt{{(4 - 2)}^{2} + {(12 - 2)}^{2} + {(5 - 1)}^{2}}$ $= 10.954451$
6. Midpoint M: (5.5, 6.5, 3.5)
1. The midpoint M is halfway, so average the matching coordinates of the two ends A and B
  $M = ((xA + xB)/2, (yA + yB)/2, (zA + zB)/2)$
2. Add the two x-coordinates and divide by 2
  $xM$ $= \frac{2 + 9}{2}$ $= 5.5$
3. Add the two y-coordinates and divide by 2
  $yM$ $= \frac{3 + 10}{2}$ $= 6.5$
4. Add the two z-coordinates and divide by 2
  $zM$ $= \frac{1 + 6}{2}$ $= 3.5$
5. Put the coordinates together to get the point M
  $M = (5.5, 6.5, 3.5)$
6. For reference, find the length of AB: square each change, add them and take the square root
  $AB$ $= \sqrt{{(9 - 2)}^{2} + {(10 - 3)}^{2} + {(6 - 1)}^{2}}$ $= 11.090537$
7. Missing endpoint B: (6, 12, 5)
1. M is halfway, so the missing end B sits as far past M as A sits before it. Double each M number and subtract A's
  $B = (2xM − xA, 2yM − yA, 2zM − zA)$
2. Find the x-coordinate of B: twice M's, minus A's
  $xB$ $= 2 \times 4 - 2$ $= 6$
3. Find the y-coordinate of B: twice M's, minus A's
  $yB$ $= 2 \times 8 - 4$ $= 12$
4. Find the z-coordinate of B: twice M's, minus A's
  $zB$ $= 2 \times 3 - 1$ $= 5$
5. Put the coordinates together to get the point B
  $B = (6, 12, 5)$
6. For reference, find the length of AB: square each change, add them and take the square root
  $AB$ $= \sqrt{{(6 - 2)}^{2} + {(12 - 4)}^{2} + {(5 - 1)}^{2}}$ $= 9.797959$
8. Point P dividing AB in the ratio 1 : 2: (4, 4.333333, 2)
1. Find t, the fraction of the way from A to B where P sits. It is m divided by the total of m and n
  $t$ $= \frac{1}{1 + 2}$ $= 0.333333$
2. Find the x-coordinate of P: start at A's and add t times the change from A to B
  $xP$ $= 2 + 0.333333 \times (8 - 2)$ $= 4$
3. Find the y-coordinate of P: start at A's and add t times the change from A to B
  $yP$ $= 2 + 0.333333 \times (9 - 2)$ $= 4.333333$
4. Find the z-coordinate of P: start at A's and add t times the change from A to B
  $zP$ $= 1 + 0.333333 \times (4 - 1)$ $= 2$
5. Put the coordinates together to get the point P
  $P = (4, 4.333333, 2)$
6. For reference, find the length of AB: square each change, add them and take the square root
  $AB$ $= \sqrt{{(8 - 2)}^{2} + {(9 - 2)}^{2} + {(4 - 1)}^{2}}$ $= 9.69536$
9. Missing endpoint B: (8, 9)
1. M is halfway, so the missing end B sits as far past M as A sits before it. Double each M number and subtract A's
  $B = (2xM − xA, 2yM − yA)$
2. Find the x-coordinate of B: twice M's, minus A's
  $xB$ $= 2 \times 5 - 2$ $= 8$
3. Find the y-coordinate of B: twice M's, minus A's
  $yB$ $= 2 \times 6 - 3$ $= 9$
4. Put the coordinates together to get the point B
  $B = (8, 9)$
5. For reference, find the length of AB: square each change, add them and take the square root
  $AB$ $= \sqrt{{(8 - 2)}^{2} + {(9 - 3)}^{2}}$ $= 8.485281$
10. Missing endpoint B: (10, 13, 5)
1. M is halfway, so the missing end B sits as far past M as A sits before it. Double each M number and subtract A's
  $B = (2xM − xA, 2yM − yA, 2zM − zA)$
2. Find the x-coordinate of B: twice M's, minus A's
  $xB$ $= 2 \times 6 - 2$ $= 10$
3. Find the y-coordinate of B: twice M's, minus A's
  $yB$ $= 2 \times 8 - 3$ $= 13$
4. Find the z-coordinate of B: twice M's, minus A's
  $zB$ $= 2 \times 3 - 1$ $= 5$
5. Put the coordinates together to get the point B
  $B = (10, 13, 5)$
6. For reference, find the length of AB: square each change, add them and take the square root
  $AB$ $= \sqrt{{(10 - 2)}^{2} + {(13 - 3)}^{2} + {(5 - 1)}^{2}}$ $= 13.416408$