The **midpoint** is the halfway point between 2 endpoints of a line segment. Certain line segments, such as medians and right bisectors are found using the midpoint. It can be found using the following formula:

\(\text{midpoint} = (\cfrac{x₁ + x₂}{2}, \cfrac{y₁ + y₂}{2})\)
- \(x₁\) represents the x-coordinate of the first point
- \(x₂\) represents the x-coordinate of the second point
- \(y₁\) represents the y-coordinate of the first point
- y₂ represents the y-coordinate of the second point

**Example**
Find the midpoint of a line segment with endpoints \(\text{C}(-4, 3)\) and \(\text{D}(2, -5)\).

We can plug the coordinates into the formula to determine the midpoint. In this instance, \(\text{C}\) will represent the first coordinate and \(\text{D}\) will represent the second coordinate:

midpoint_{CD} \(= (\cfrac{-4 + 2}{2}, \cfrac{3 + (-5)}{2})\)

midpoint_{CD} \(= (\cfrac{-2}{2}, \cfrac{-2}{2})\)

midpoint_{CD} \(= (-1, -1)\)
Therefore, we can determine that the midpoint between points \(\text{C}\) and \(\text{D}\) is \((-1, -1)\).

Find the midpoint of a line segment with endpoints \(\text{A}(-2/5, -3/4)\) and \(\text{B}(4/5, 3/4)\).

Show Answer
First, we need to ensure that all fractions have the same denominator so that the formula works properly. In this case, all fractions will have a denominator of \(20\):

\(\text{A}(\cfrac{-8}{20}, \cfrac{-15}{20}), \text{B}(\cfrac{16}{20}, \cfrac{15}{20})\)
Next, we can plug the coordinates into the formula to find the midpoint. In this instance, \(\text{A}\) will represent the first coordinate and \(\text{B}\) will represent the second coordinate:

midpoint_{AB} \(= (\cfrac{\cfrac{-8}{20} + \cfrac{16}{20}}{2}, \cfrac{\cfrac{-15}{20} + \cfrac{15}{20}}{2})\)

midpoint_{AB} \(= (\cfrac{\cfrac{8}{20}}{2}, \cfrac{\cfrac{0}{20}}{2})\)

midpoint_{AB} \(= (\cfrac{4}{20}, 0) =\) \((\cfrac{1}{5}, 0)\)
Therefore, we can determine that the midpoint between points \(\text{A}\) and \(\text{B}\) is \((\cfrac{1}{5}, 0)\).

The endpoints of the diameter of a circle are \(\text{A}(-5, -3)\) and \(\text{B}(3, 7)\). Find the coordinates of the circle's origin.

Show Answer
We can use the midpoint formula to determine the coordinates of the circle's origin. In this instance, \(\text{A}\) represents coordinate \(1\) and \(\text{B}\) represents coordinate \(2\):

\(\text{origin} = (\cfrac{-5 + 3}{2}, \cfrac{-3 + 7}{2})\)

\(\text{origin} = (\cfrac{-2}{2}, \cfrac{4}{2})\)

\(\text{origin} = (-1, 2)\)

Therefore, we can determine that the coordinates for the circle's origin are \((-1, 2)\).

1.Draw △ABC with vertices \(\text{A}(-8, 0)\), \(\text{B}(0, 0)\) and \(\text{C}(0, -8)\).

2. Construct the midpoints \(\text{AB}\), \(\text{BC}\), and \(\text{AC}\) and label them \(\text{D}\), \(\text{E}\), and \(\text{F}\) respectively. Join the midpoints to form △\(\text{DEF}\).

3. Show that line segment \(\text{DE}\) is parallel to the line segment \(\text{AC}\).

Show Answer
i. Based on the set of coordinates given in the question, we can draw the triangle:

ii. We can construct the midpoints using the midpoint formula:

midpoint_{AB} \(= (\cfrac{-8 + 0}{2}, \cfrac{0 + 0}{2})\)

midpoint_{AB} \(= (\cfrac{-8}{2}, \cfrac{0}{2})\)

midpoint_{AB} \(= (-4, 0)\)

midpoint_{BC} \(= (\cfrac{0 + 0}{2}, \cfrac{0 - 8}{2})\)

midpoint_{BC} \(= (\cfrac{0}{2}, \cfrac{-8}{2})\)

midpoint_{BC} \(= (0, -4)\)

midpoint_{AC} \(= (\cfrac{-8 + 0}{2}, \cfrac{0 - 8}{2})\)

midpoint_{AC} \(= (\cfrac{-8}{2}, \cfrac{-8}{2})\)

midpoint_{AC} \(= (-4, -4)\)

Using these coordinates, we can now graph △\(\text{DEF}\):

iii. To show that lines \(\text{AC}\) and \(\text{DE}\) are parallel to one another, we can compare their slopes:

\(\text{AC} = (\cfrac{-8 - 0}{0 - (-8)})\)

\(\text{AC} = (\cfrac{-8}{8}) = -1\)

\(\text{DE} = (\cfrac{-4 - 0}{0 - (-4)})\)

\(\text{DE} = (\cfrac{-4}{4}) = -1\)

\(\text{AC} = \text{DE}\)
As lines \(\text{AC}\) and \(\text{DE}\) have the same slope, we can confirm that they are parallel to each other.