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Worksheet: Midpoint of a Line Segment

Q1:

Given 𝐴 ( 4 , 8 ) and 𝐡 ( 6 , 6 ) , what are the coordinates of the midpoint of 𝐴 𝐡 ?

  • A ( 7 , 5 )
  • B ( βˆ’ 1 , 1 )
  • C ( 6 , 6 )
  • D ( 5 , 7 )

Q2:

Given 𝐴 ( 2 , 1 ) and 𝐢 ( βˆ’ 8 , βˆ’ 9 ) , what are the coordinates of 𝐡 , if 𝐢 is the midpoint of 𝐴 𝐡 ?

  • A ( 1 0 , 1 0 )
  • B ( βˆ’ 4 , βˆ’ 7 )
  • C ( βˆ’ 3 , βˆ’ 4 )
  • D ( 1 2 , 1 1 )

Q3:

Given 𝐴 ( 0 , 2 ) and 𝐢 ( βˆ’ 2 , 0 ) , what are the coordinates of 𝐡 , if 𝐢 is the midpoint of 𝐴 𝐡 ?

  • A ( 2 , 2 )
  • B ( βˆ’ 2 , 4 )
  • C ( βˆ’ 1 , 1 )
  • D ( 2 , 4 )

Q4:

A rectangular yard is next to a house along a road. In the yard is an orange tree 7 m from the house and 3 m from the road. There is also an apple tree, 5 m from the house and 9 m from the road. A fountain is placed halfway between the trees. How far is the fountain from the house and the road?

  • A 1 m, 3 m
  • B 3.5 m, 4.5 m
  • C 6 m, 3.5 m
  • D 6 m, 6 m
  • E 4 m, 8 m

Q5:

Find the point 𝐴 on the π‘₯ -axis and the point 𝐡 on the 𝑦 -axis such that ο€Ό 3 2 , βˆ’ 5 2  is the midpoint of 𝐴 𝐡 .

  • A ( 3 , 3 ) , ( βˆ’ 5 , βˆ’ 5 )
  • B ( 0 , 3 ) , ( βˆ’ 5 , 0 )
  • C ( 3 , βˆ’ 5 ) , ( βˆ’ 5 , 3 )
  • D ( 3 , 0 ) , ( 0 , βˆ’ 5 )
  • E ( 0 , βˆ’ 5 ) , ( 3 , 0 )

Q6:

Find the point 𝐴 on the π‘₯ -axis and the point 𝐡 on the 𝑦 -axis such that ο€Ό 9 2 , βˆ’ 7 2  is the midpoint of 𝐴 𝐡 .

  • A ( 9 , 9 ) , ( βˆ’ 7 , βˆ’ 7 )
  • B ( 0 , 9 ) , ( βˆ’ 7 , 0 )
  • C ( 9 , βˆ’ 7 ) , ( βˆ’ 7 , 9 )
  • D ( 9 , 0 ) , ( 0 , βˆ’ 7 )
  • E ( 0 , βˆ’ 7 ) , ( 9 , 0 )

Q7:

Point ( 2 , βˆ’ 7 ) is the midpoint of the line segment on endpoints ( π‘₯ , βˆ’ 9 ) and ( 1 , 𝑦 ) . What is π‘₯ + 𝑦 ?

Q8:

The coordinates of the points 𝐴 , 𝐡 , and 𝐢 in the parallelogram 𝐴 𝐡 𝐢 𝐷 are ( βˆ’ 2 , βˆ’ 5 ) , ( βˆ’ 5 , βˆ’ 7 ) , and ( βˆ’ 1 , βˆ’ 1 3 ) , respectively. If the point 𝐸 lies on  𝐴 𝐷 such that 𝐴 𝐸 = 2 𝐴 𝐷 , determine the coordinates of the points 𝐷 and 𝐸 .

  • A 𝐷 ( βˆ’ 6 , 1 ) , 𝐸 ( βˆ’ 1 0 , 7 )
  • B 𝐷 ( 3 , βˆ’ 1 9 ) , 𝐸 ( 7 , βˆ’ 2 5 )
  • C 𝐷 ( 2 , βˆ’ 1 1 ) , 𝐸 ( βˆ’ 2 , βˆ’ 5 )
  • D 𝐷 ( 2 , βˆ’ 1 1 ) , 𝐸 ( 6 , βˆ’ 1 7 )

Q9:

The coordinates of the points 𝐴 , 𝐡 , and 𝐢 in the parallelogram 𝐴 𝐡 𝐢 𝐷 are ( 2 , βˆ’ 6 ) , ( βˆ’ 5 , 6 ) , and ( 1 , βˆ’ 3 ) , respectively. If the point 𝐸 lies on  𝐴 𝐷 such that 𝐴 𝐸 = 2 𝐴 𝐷 , determine the coordinates of the points 𝐷 and 𝐸 .

  • A 𝐷 ( βˆ’ 4 , 3 ) , 𝐸 ( βˆ’ 1 0 , 1 2 )
  • B 𝐷 ( 7 , βˆ’ 1 2 ) , 𝐸 ( 1 3 , βˆ’ 2 1 )
  • C 𝐷 ( 8 , βˆ’ 1 5 ) , 𝐸 ( 2 , βˆ’ 6 )
  • D 𝐷 ( 8 , βˆ’ 1 5 ) , 𝐸 ( 1 4 , βˆ’ 2 4 )

Q10:

On the graph, which point is halfway between ( 1 , 8 ) and ( 5 , 2 ) ?

  • A ( 2 , 3 )
  • B ( 5 , 3 )
  • C ( 3 , 2 )
  • D ( 3 , 5 )
  • E ( 4 , 6 )

Q11:

Consider the two points 𝐴 ( π‘₯ , 𝑦 )   and 𝐡 ( π‘₯ , 𝑦 )   .

Find an expression for the midpoint of the line segment 𝐴 𝐡 .

  • A ( π‘₯ + π‘₯ , 𝑦 + 𝑦 )    
  • B ο€» π‘₯ βˆ’ π‘₯ 2 , 𝑦 βˆ’ 𝑦 2     
  • C ( π‘₯ βˆ’ π‘₯ , 𝑦 βˆ’ 𝑦 )    
  • D ο€Ό π‘₯ + π‘₯ 2 , 𝑦 + 𝑦 2     
  • E ( 2 ( π‘₯ + π‘₯ ) , 2 ( 𝑦 + 𝑦 ) )    

𝐴 and 𝐡 have the coordinates ( 1 , 1 ) and ( 3 , βˆ’ 5 ) respectively. Find the midpoint of the line segment 𝐴 𝐡 .

  • A ( 2 , βˆ’ 2 )
  • B(0, 6)
  • C ( 4 , βˆ’ 4 )
  • D(0, 3)
  • E ( 8 , βˆ’ 8 )

Q12:

Points 𝐴 and 𝐡 have coordinates ( 3 , 3 ) and ( βˆ’ 2 , βˆ’ 5 ) respectively. Is the point ο€Ό 1 2 , βˆ’ 1  the midpoint of segment 𝐴 𝐡 ?

  • A We need more information in order to work it out.
  • B no
  • C yes

Q13:

Let 𝐴 ( βˆ’ 3 , βˆ’ 8 ) , 𝐡 ( βˆ’ 3 , 6 ) , and 𝐢 ( 3 , βˆ’ 8 ) be the vertices of a triangle. Suppose that point 𝐷 on 𝐡 𝐢 is such that  𝐴 𝐷 bisects the angle at 𝐴 . What is the point 𝐷 ?

  • A ο€Ό βˆ’ 6 5 , 9 5 
  • B ( 0 , βˆ’ 1 )
  • C ο€Ό 1 2 7 , βˆ’ 3 8 7 
  • D ο€Ό 6 5 , βˆ’ 1 9 5 

Q14:

Find the values of π‘Ž and 𝑏 so that ( βˆ’ 2 π‘Ž , 2 π‘Ž + 𝑏 ) is the midpoint of the line segment between ( βˆ’ 2 , βˆ’ 3 ) and ( 2 , 1 1 ) .

  • A π‘Ž = 2 , 𝑏 = βˆ’ 1 8
  • B π‘Ž = βˆ’ 2 , 𝑏 = 1 8
  • C π‘Ž = 0 , 𝑏 = 8
  • D π‘Ž = 0 , 𝑏 = 4
  • E π‘Ž = 1 , 𝑏 = βˆ’ 9

Q15:

Find the values of π‘Ž and 𝑏 so that ( π‘Ž , βˆ’ 2 π‘Ž βˆ’ 2 𝑏 ) is the midpoint of the line segment between ( βˆ’ 6 , βˆ’ 2 ) and ( 6 , 1 0 ) .

  • A π‘Ž = βˆ’ 1 2 , 𝑏 = 1 8
  • B π‘Ž = 1 2 , 𝑏 = βˆ’ 1 8
  • C π‘Ž = 0 , 𝑏 = βˆ’ 4
  • D π‘Ž = 0 , 𝑏 = βˆ’ 2
  • E π‘Ž = βˆ’ 6 , 𝑏 = 9

Q16:

𝐢 is the midpoint of 𝐴 𝐡 . Find the values of π‘₯ and 𝑦 if the coordinates of 𝐴 , 𝐡 , and 𝐢 are ( π‘₯ , 4 ) , ( 3 , βˆ’ 2 ) , and ( 9 , 𝑦 ) respectively.

  • A π‘₯ = 1 , 𝑦 = 6
  • B π‘₯ = 2 1 , 𝑦 = 3
  • C π‘₯ = 6 , 𝑦 = 1
  • D π‘₯ = 1 5 , 𝑦 = 1

Q17:

The point 𝐢 is on ray οƒ  𝐴 𝐡 but not segment 𝐴 𝐡 , and its distance from 𝐴 ( 3 , 0 ) is 2 times its distance from 𝐡 ( βˆ’ 9 , βˆ’ 6 ) . What are its coordinates?

  • A ( 1 5 , 6 )
  • B ( βˆ’ 5 , βˆ’ 4 )
  • C ( βˆ’ 1 , βˆ’ 2 )
  • D ( βˆ’ 2 1 , βˆ’ 1 2 )

Q18:

Suppose the circle of center 𝑀 ο€Ό βˆ’ 4 , 1 2  and diameter 𝐴 𝐡 , where 𝐡 ( βˆ’ 3 , 0 ) . Find the coordinates of 𝐴 , and give the circumference to two decimal places.

  • A 𝐴 ( βˆ’ 1 1 , 1 ) , 3.93
  • B 𝐴 ( 5 , βˆ’ 1 ) , 3.51
  • C 𝐴 ( 1 1 , βˆ’ 1 ) , 7.02
  • D 𝐴 ( βˆ’ 5 , 1 ) , 7.02

Q19:

Consider the points 𝐴 ( 7 , 7 ) , 𝐡 ( 9 , βˆ’ 7 ) , and 𝐢 ( 5 , 1 ) . Given that 𝐴 𝐷 is a median of the triangle 𝐴 𝐡 𝐢 and 𝑀 is the midpoint of this median, determine the coordinates of 𝐷 and 𝑀 .

  • A ( 1 , 3 ) , ( βˆ’ 1 , 2 )
  • B ( 2 , βˆ’ 4 ) , ( 0 , βˆ’ 5 )
  • C ( 3 , 1 ) , ( βˆ’ 5 , 0 )
  • D ( 7 , βˆ’ 3 ) , ( 7 , 2 )

Q20:

The origin is the midpoint of the straight segment 𝐴 𝐡 . Find the coordinates of point 𝐡 if the coordinates of point 𝐴 are ( βˆ’ 6 , 4 ) .

  • A ( βˆ’ 4 , 6 )
  • B ( βˆ’ 6 , 4 )
  • C ( βˆ’ 3 , 2 )
  • D ( 6 , βˆ’ 4 )

Q21:

𝐴 , 𝐡 , 𝐢 , and 𝐷 are collinear points. Suppose that the coordinates of points 𝐴 and 𝐢 are ( 2 , 4 ) and ( βˆ’ 8 , βˆ’ 8 ) , respectively, and that 𝐴 𝐡 = 𝐡 𝐢 = 𝐢 𝐷 . What are the coordinates of 𝐡 and 𝐷 ?

  • A ( βˆ’ 1 , 0 ) , ( βˆ’ 7 , βˆ’ 8 )
  • B ( 3 , βˆ’ 8 ) , ( βˆ’ 1 3 , βˆ’ 2 4 )
  • C ( 5 , 6 ) , ( βˆ’ 2 1 , βˆ’ 2 2 )
  • D ( βˆ’ 3 , βˆ’ 2 ) , ( βˆ’ 1 3 , βˆ’ 1 4 )

Q22:

𝐴 , 𝐡 , 𝐢 , and 𝐷 are collinear points. Suppose that the coordinates of points 𝐴 and 𝐢 are ( 4 , βˆ’ 2 ) and ( βˆ’ 4 , 0 ) , respectively, and that 𝐴 𝐡 = 𝐡 𝐢 = 𝐢 𝐷 . What are the coordinates of 𝐡 and 𝐷 ?

  • A ( 3 , βˆ’ 2 ) , ( βˆ’ 7 , 2 )
  • B ( 1 , βˆ’ 2 ) , ( βˆ’ 7 , βˆ’ 2 )
  • C ( 4 , βˆ’ 1 ) , ( βˆ’ 4 , βˆ’ 7 )
  • D ( 0 , βˆ’ 1 ) , ( βˆ’ 8 , 1 )

Q23:

𝐴 ( 3 , 1 7 ) , 𝐹 ( 1 0 , 1 7 ) , 𝐡 ( 1 7 , 1 7 ) , 𝐷 ( 4 , 8 . 5 ) , and 𝐢 ( 1 8 . 5 , 4 . 2 5 ) are points on the trapezoid 𝐴 𝐡 𝐢 𝐷 . If 𝐹 𝐺 is parallel to 𝐴 𝐷 , what is the π‘₯ -coordinate of point 𝐺 ?

Q24:

If 𝐢 ( βˆ’ 5 , 4 ) is the midpoint of οƒ  𝐴 𝐡 , where 𝐴 ( π‘₯ , 4 ) , 𝐡 ( βˆ’ 5 , 𝑦 ) , find the values of π‘₯ and 𝑦 .

  • A π‘₯ = βˆ’ 1 5 , 𝑦 = 1 2
  • B π‘₯ = 4 , 𝑦 = βˆ’ 5
  • C π‘₯ = βˆ’ 1 0 , 𝑦 = 8
  • D π‘₯ = βˆ’ 5 , 𝑦 = 4

Q25:

Consider the points 𝐴 ( π‘₯ , 7 ) , 𝐡 ( βˆ’ 4 , 𝑦 ) , and 𝐢 ( 2 , 5 ) . Given that 𝐢 is the midpoint of 𝐴 𝐡 , find the values of π‘₯ and 𝑦 .

  • A π‘₯ = βˆ’ 1 , 𝑦 = 6
  • B π‘₯ = 0 , 𝑦 = 1 7
  • C π‘₯ = 6 , 𝑦 = βˆ’ 2
  • D π‘₯ = 8 , 𝑦 = 3