Midpoint & Distance Formula Answer Key: A worksheet with four geometry problems involving midpoints, medians, and circle properties, each with a diagram and solution.

Let’s solve each problem step by step.

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Problem 1:

We are told that D is the midpoint of BC, and we need to find the length of median AD.

Points given:
- A(-4, 4)
- B(-1, -2)
- C(-7, -2)

First, find coordinates of D (midpoint of BC).

Midpoint formula:
D = ((x₁ + x₂)/2 , (y₁ + y₂)/2)

So for B(-1, -2) and C(-7, -2):

x-coordinate of D = (-1 + (-7)) / 2 = (-8)/2 = -4
y-coordinate of D = (-2 + (-2)) / 2 = (-4)/2 = -2

So D is at (-4, -2)

Now find distance between A(-4, 4) and D(-4, -2)

Distance formula:
√[(x₂ - x₁)² + (y₂ - y₁)²]

Plug in:

= √[(-4 - (-4))² + (-2 - 4)²]
= √[(0)² + (-6)²]
= √[0 + 36]
= √36 = 6

✔ So length of AD is 6 units

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Problem 2:

X and Y are midpoints of RS and PQ. Find length XY.

Points:
- R(7, -3), S(3, -3) → so X is midpoint of RS
- P(1, -7), Q(9, -7) → so Y is midpoint of PQ

Find X:

X = ((7 + 3)/2 , (-3 + (-3))/2) = (10/2, -6/2) = (5, -3)

Find Y:

Y = ((1 + 9)/2 , (-7 + (-7))/2) = (10/2, -14/2) = (5, -7)

Now find distance between X(5, -3) and Y(5, -7)

Distance = √[(5 - 5)² + (-7 - (-3))²]
= √[0 + (-4)²]
= √16 = 4

✔ So length of XY is 4 units

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Problem 3:

Z is the midpoint of EG. Find length of FZ.

Points:
- E(7, 5), G(3, 7) → Z is midpoint
- F(8, 7)

First, find Z:

Z = ((7 + 3)/2 , (5 + 7)/2) = (10/2, 12/2) = (5, 6)

Now find distance between F(8, 7) and Z(5, 6)

Distance = √[(8 - 5)² + (7 - 6)²]
= √[3² + 1²]
= √[9 + 1] = √10 ≈ 3.162...

Rounded to two decimal places: 3.16

✔ So length of FZ is 3.16 units

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Problem 4:

O is the center of the circle. M(-8, -5) and N(2, -5) are endpoints of a diameter (since O is center and they’re on opposite sides).

Center O is midpoint of MN.

Midpoint O = ((-8 + 2)/2 , (-5 + (-5))/2) = (-6/2, -10/2) = (-3, -5)

Radius is half the diameter, or distance from center to either end.

Let’s compute distance from O(-3, -5) to N(2, -5):

Distance = √[(2 - (-3))² + (-5 - (-5))²]
= √[(5)² + 0] = √25 = 5

✔ So center is (-3, -5) and radius is 5 units

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Final Answer:
1) 6 units
2) 4 units
3) 3.16 units
4) Center: (-3, -5); Radius: 5 units