Solved Exercises on Midpoint Formula in Grade 8

Master midpoint formula: finding midpoints, applications, and problem-solving through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Midpoint Formula
Exercise 1
Find the midpoint of the line segment joining points A(2, 5) and B(8, 11).
Definition:

Midpoint formula: The midpoint of a line segment joining points (x₁, y₁) and (x₂, y₂) is M = ((x₁+x₂)/2, (y₁+y₂)/2)

Solution method:
  1. Identify coordinates: (x₁, y₁) and (x₂, y₂)
  2. Substitute into midpoint formula
  3. Calculate average of x-coordinates
  4. Calculate average of y-coordinates
Given
A(2,5), B(8,11)
Formula
M = ((2+8)/2, (5+11)/2)
Solution
M = (5, 8)
Step 1: Identify coordinates

Point A: (x₁, y₁) = (2, 5)

Point B: (x₂, y₂) = (8, 11)

Step 2: Apply midpoint formula

M = ((x₁+x₂)/2, (y₁+y₂)/2)

M = ((2+8)/2, (5+11)/2)

Step 3: Calculate x-coordinate

x-coordinate = (2+8)/2 = 10/2 = 5

Step 4: Calculate y-coordinate

y-coordinate = (5+11)/2 = 16/2 = 8

Midpoint = (5, 8)
Final answer:

The midpoint of the line segment joining A(2, 5) and B(8, 11) is (5, 8)

Applied rules:

• Midpoint formula: Average of x-coordinates and y-coordinates

• Order doesn't matter: (x₁+x₂)/2 = (x₂+x₁)/2

• Verification: Check that (5, 8) is between (2, 5) and (8, 11)

2 Midpoint with Negative Coordinates
Exercise 2
Find the midpoint of the line segment joining points P(-4, 3) and Q(6, -7).
Definition:

Handling negative coordinates: The midpoint formula works the same with negative values

Given
P(-4,3), Q(6,-7)
Formula
M = ((-4+6)/2, (3+(-7))/2)
Solution
M = (1, -2)
Step 1: Identify coordinates

Point P: (x₁, y₁) = (-4, 3)

Point Q: (x₂, y₂) = (6, -7)

Step 2: Apply midpoint formula

M = ((x₁+x₂)/2, (y₁+y₂)/2)

M = ((-4+6)/2, (3+(-7))/2)

Step 3: Calculate x-coordinate

x-coordinate = (-4+6)/2 = 2/2 = 1

Step 4: Calculate y-coordinate

y-coordinate = (3+(-7))/2 = -4/2 = -2

Midpoint = (1, -2)
Final answer:

The midpoint of the line segment joining P(-4, 3) and Q(6, -7) is (1, -2)

Applied rules:

• Negative addition: Adding a negative is subtracting: -4 + 6 = 2

• Division with negatives: (-4)/2 = -2

• Midpoint location: Lies between the original points

3 Real-World Application
Exercise 3
On a map, City A is located at (-2, 4) and City B is located at (10, 12). A rest stop is built exactly halfway between the cities. Where is the rest stop located?
Definition:

Real-world midpoint: The midpoint represents the halfway point between two locations

Given
City A(-2,4), City B(10,12)
Formula
M = ((-2+10)/2, (4+12)/2)
Solution
M = (4, 8)
Step 1: Identify coordinates

City A: (x₁, y₁) = (-2, 4)

City B: (x₂, y₂) = (10, 12)

Step 2: Apply midpoint formula

M = ((x₁+x₂)/2, (y₁+y₂)/2)

M = ((-2+10)/2, (4+12)/2)

Step 3: Calculate x-coordinate

x-coordinate = (-2+10)/2 = 8/2 = 4

Step 4: Calculate y-coordinate

y-coordinate = (4+12)/2 = 16/2 = 8

Step 5: Interpret the result

The rest stop is located at coordinates (4, 8)

Rest stop at (4, 8)
Final answer:

The rest stop is located at coordinates (4, 8)

Applied rules:

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

• Real-world application: Midpoint represents the halfway point

• Verification: Check that (4, 8) is between (-2, 4) and (10, 12)

Rules and methods, laws,...
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Midpoint Formula
M = (x₁/2, y₁/2)
Midpoint from Origin
(x₁+x₂)/2 = x_m
X-Coordinate Only
Horizontal Segment
M = ((x₁+x₂)/2, y)
When y₁ = y₂
Vertical Segment
M = (x, (y₁+y₂)/2)
When x₁ = x₂
Number Line
M = (x₁+x₂)/2
One-dimensional midpoint
Key definitions:

Midpoint: The point that divides a line segment into two equal parts

Line segment: A part of a line bounded by two endpoints

Midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2) for endpoints (x₁, y₁) and (x₂, y₂)

Coordinate plane: A two-dimensional plane defined by x-axis and y-axis

Ordered pair: A point represented as (x, y) where x is horizontal and y is vertical

Endpoint: Either of the two points that define a line segment

Equal segments: The midpoint creates two line segments of equal length

Complete methodology:
  1. Identify the endpoints: Determine (x₁, y₁) and (x₂, y₂) for the line segment
  2. Substitute into formula: Place coordinates into M = ((x₁+x₂)/2, (y₁+y₂)/2)
  3. Calculate x-coordinate: Find (x₁+x₂)/2
  4. Calculate y-coordinate: Find (y₁+y₂)/2
  5. Express the midpoint: Write as an ordered pair (x_m, y_m)
  6. Verify the result: Check that the midpoint lies between the original points
Tip 1: The midpoint is always between the two endpoints.
Tip 2: Order doesn't matter: (x₁+x₂)/2 = (x₂+x₁)/2.
Tip 3: For horizontal segments, only the x-coordinate changes.
Tip 4: For vertical segments, only the y-coordinate changes.
Common errors: Forgetting to divide by 2, mixing up x and y coordinates, making arithmetic errors with negative numbers, confusing with distance formula.
Exam preparation: Practice with various coordinate combinations, memorize the formula, work on problems with different contexts, understand the geometric meaning.
Formulas to know by heart:

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

• Horizontal segment: M = ((x₁+x₂)/2, y) when y₁ = y₂

• Vertical segment: M = (x, (y₁+y₂)/2) when x₁ = x₂

• From origin: M = (x₁/2, y₁/2) for segment from (0,0) to (x₁,y₁)

• Number line: M = (x₁+x₂)/2

• Relationship to distance: Midpoint is equidistant from both endpoints

Solution: Exercises 4 to 5
4 Finding Endpoint Given Midpoint
Exercise 4
The midpoint of a line segment is (3, -1) and one endpoint is (-1, 4). Find the other endpoint.
Definition:

Reversing midpoint formula: If M = ((x₁+x₂)/2, (y₁+y₂)/2), then (x₂, y₂) = (2x_m - x₁, 2y_m - y₁)

Given
M(3,-1), A(-1,4)
Formula
B = (2×3-(-1), 2×(-1)-4)
Solution
B = (7, -6)
Step 1: Identify known values

Midpoint: M = (x_m, y_m) = (3, -1)

One endpoint: A = (x₁, y₁) = (-1, 4)

Unknown endpoint: B = (x₂, y₂) = ?

Step 2: Use midpoint formula to derive equations

From M = ((x₁+x₂)/2, (y₁+y₂)/2):

3 = (-1+x₂)/2

-1 = (4+y₂)/2

Step 3: Solve for x₂

3 = (-1+x₂)/2

6 = -1+x₂

x₂ = 7

Step 4: Solve for y₂

-1 = (4+y₂)/2

-2 = 4+y₂

y₂ = -6

Step 5: Verify the result

Midpoint of (-1, 4) and (7, -6):

((−1+7)/2, (4+(−6))/2) = (6/2, -2/2) = (3, -1) ✓

Other endpoint = (7, -6)
Final answer:

The other endpoint is (7, -6)

Applied rules:

• Algebraic manipulation: Solve midpoint formula for unknown endpoint

• Verification: Check that the calculated point gives the correct midpoint

• General formula: If M is midpoint of A and B, then B = (2M_x - A_x, 2M_y - A_y)

5 Geometric Applications
Exercise 5
Show that the diagonals of quadrilateral ABCD with vertices A(0, 0), B(4, 0), C(4, 3), and D(0, 3) bisect each other by finding the midpoints of both diagonals.
Definition:

Diagonal bisection: When diagonals of a quadrilateral intersect at their midpoints

Diagonal AC
((0+4)/2, (0+3)/2) = (2, 1.5)
Diagonal BD
((4+0)/2, (0+3)/2) = (2, 1.5)
Same point
✓ Bisect each other
Step 1: Identify the diagonals

Diagonal AC connects A(0, 0) and C(4, 3)

Diagonal BD connects B(4, 0) and D(0, 3)

Step 2: Find midpoint of diagonal AC

M₁ = ((0+4)/2, (0+3)/2)

M₁ = (4/2, 3/2) = (2, 1.5)

Step 3: Find midpoint of diagonal BD

M₂ = ((4+0)/2, (0+3)/2)

M₂ = (4/2, 3/2) = (2, 1.5)

Step 4: Compare the midpoints

M₁ = (2, 1.5) and M₂ = (2, 1.5)

Since both midpoints are the same, the diagonals bisect each other

Diagonals bisect at (2, 1.5)
Final answer:

The diagonals bisect each other at the point (2, 1.5), confirming that ABCD is a parallelogram.

Applied rules:

• Midpoint formula: Calculate midpoint of each diagonal

• Parallelogram property: Diagonals bisect each other

• Verification: Equal midpoints prove bisection

Key Concepts, Laws, Methods, and Formulas for Midpoint Formula
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Midpoint Formula
Key definitions:

Midpoint: The point that divides a line segment into two equal parts, located exactly halfway between the endpoints

Line segment: A part of a line bounded by two distinct endpoints

Midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2), where (x₁, y₁) and (x₂, y₂) are the endpoints

Coordinate plane: A two-dimensional plane with x-axis (horizontal) and y-axis (vertical) intersecting at the origin (0, 0)

Ordered pair: A pair of numbers (x, y) that describes the position of a point in the coordinate plane

Endpoint: Either of the two points that define a line segment

Bisect: To divide into two equal parts

Geometric mean: The midpoint represents the average position of the endpoints

Segment partition: The midpoint creates two congruent segments

Complete methodology:
  1. Identify the endpoints: Carefully read and record the coordinates of both endpoints of the line segment
  2. Assign coordinates: Designate one endpoint as (x₁, y₁) and the other as (x₂, y₂)
  3. Substitute into formula: Place the coordinates into M = ((x₁+x₂)/2, (y₁+y₂)/2)
  4. Calculate x-coordinate: Find (x₁+x₂)/2, the average of the x-coordinates
  5. Calculate y-coordinate: Find (y₁+y₂)/2, the average of the y-coordinates
  6. Express the result: Write the midpoint as an ordered pair (x_m, y_m)
  7. Verify the result: Check that the midpoint lies between the original points and appears reasonable
  8. Apply context: Interpret the result in the context of the problem if applicable
Tip 1: The midpoint is always between the two endpoints - it should have an x-value between the two x-values and a y-value between the two y-values.
Tip 2: Order doesn't matter since addition is commutative: (x₁+x₂)/2 = (x₂+x₁)/2.
Tip 3: For horizontal line segments (same y-values), only the x-coordinate changes in the midpoint.
Tip 4: For vertical line segments (same x-values), only the y-coordinate changes in the midpoint.
Tip 5: The midpoint is equidistant from both endpoints - this can be used to verify your answer.
Tip 6: When solving for an unknown endpoint given the midpoint and one endpoint, use algebra to rearrange the midpoint formula.
Common errors: Forgetting to divide by 2, mixing up x and y coordinates, making arithmetic errors with negative numbers, confusing with distance formula, not properly identifying endpoints.
Memory aids: "Average of x's, average of y's", "Add coordinates, divide by two", "The middle of the middle".
Problem-solving strategies: Plot the points to visualize the segment, check that your answer is reasonable, verify by calculating distances to endpoints, use the formula consistently.
Essential formulas and theorems:

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

• Horizontal segment: M = ((x₁+x₂)/2, y) when y₁ = y₂

• Vertical segment: M = (x, (y₁+y₂)/2) when x₁ = x₂

• From origin: M = (x₁/2, y₁/2) for segment from (0,0) to (x₁,y₁)

• Number line: M = (x₁+x₂)/2

• Reverse formula: If M is midpoint of A and B, then B = (2M_x - A_x, 2M_y - A_y)

• Distance property: Midpoint is equidistant from both endpoints

• Geometric property: Midpoint creates two congruent segments

Visual Representation: Midpoint Relationships
Exercise 6: Midpoint vs Endpoint Distances
Visual representation of how midpoint coordinates relate to endpoint coordinates:
- X-coordinate relationships
- Y-coordinate relationships
- Distance comparisons
- Geometric properties

Analysis: The chart illustrates how midpoint coordinates are averages of endpoint coordinates.

  • Midpoint x-coordinate is always between the two endpoint x-coordinates
  • Midpoint y-coordinate is always between the two endpoint y-coordinates
  • The relationship is linear (y = x/2 for distance comparisons)
  • Midpoint always creates equal segments from both endpoints

Questions & Answers

Question: I sometimes get confused about which coordinate is x and which is y in the midpoint formula. Any tips?

Answer: Here are some helpful tips to remember:

  • XY Order: Always remember "X comes before Y" alphabetically, just like in ordered pairs (x, y)
  • Horizontal-Vertical: X is horizontal (left-right), Y is vertical (up-down)
  • Formula Structure: M = ((x₁+x₂)/2, (y₁+y₂)/2) - the first fraction is for x-coordinates, the second is for y-coordinates

For example, in point A(3, 7): x = 3, y = 7

In the midpoint formula, you calculate the average of x-coordinates separately from the average of y-coordinates, then combine them as an ordered pair.

Practice writing out the formula completely: M = ((x₁+x₂)/2, (y₁+y₂)/2) before substituting values.

Question: Can the midpoint formula give fractional coordinates?

Answer: Yes, absolutely! The midpoint formula often results in fractional coordinates. This happens when:

  • The sum of x-coordinates is odd (like 3 + 4 = 7, so 7/2 = 3.5)
  • The sum of y-coordinates is odd
  • The coordinates are not both even or both odd

For example:

  • Endpoints (1, 2) and (4, 6) give midpoint ((1+4)/2, (2+6)/2) = (2.5, 4)
  • Endpoints (0, 0) and (3, 5) give midpoint ((0+3)/2, (0+5)/2) = (1.5, 2.5)

Fractional coordinates are perfectly valid and represent the exact midpoint position. You can express them as decimals or fractions.

This is different from the distance formula, which can result in irrational numbers.

Question: How can I verify that my midpoint calculation is correct?

Answer: Here are several ways to verify your midpoint calculation:

  1. Range check: The midpoint x-coordinate should be between the two endpoint x-coordinates, and the y-coordinate should be between the two endpoint y-coordinates
  2. Distance check: Calculate the distance from midpoint to each endpoint - they should be equal
  3. Reverse calculation: If M is the midpoint of A and B, then A and B should be equidistant from M
  4. Visual check: Plot the points on a coordinate plane to see if the midpoint appears to be in the center

For example, if endpoints are (2, 3) and (8, 7) and midpoint is (5, 5):

  • Range check: 2 ≤ 5 ≤ 8 and 3 ≤ 5 ≤ 7 ✓
  • Distance check: Distance from (5,5) to (2,3) = √[(5-2)² + (5-3)²] = √13
  • Distance from (5,5) to (8,7) = √[(5-8)² + (5-7)²] = √13 ✓

Question: How do I find an endpoint if I know the midpoint and the other endpoint?

Answer: You can solve this algebraically by rearranging the midpoint formula. If M is the midpoint of A and B:

Starting with: M = ((x_A + x_B)/2, (y_A + y_B)/2)

To find B when you know M and A:

  • For x-coordinate: x_M = (x_A + x_B)/2 → 2x_M = x_A + x_B → x_B = 2x_M - x_A
  • For y-coordinate: y_M = (y_A + y_B)/2 → 2y_M = y_A + y_B → y_B = 2y_M - y_A

So B = (2x_M - x_A, 2y_M - y_A)

Example: If midpoint M = (3, 4) and endpoint A = (1, 2), then:

  • x_B = 2(3) - 1 = 5
  • y_B = 2(4) - 2 = 6

So the other endpoint is B = (5, 6). Verify: ((1+5)/2, (2+6)/2) = (3, 4) ✓

Question: What's the difference between the midpoint formula and the distance formula?

Answer: These formulas serve different purposes:

Midpoint Formula:

  • Finds the point exactly halfway between two endpoints
  • Result: An ordered pair (x, y)
  • Formula: M = ((x₁+x₂)/2, (y₁+y₂)/2)
  • Uses addition and division
  • Always gives a finite, rational or irrational result

Distance Formula:

  • Finds the length of the line segment between two points
  • Result: A positive number (length)
  • Formula: d = √[(x₂-x₁)² + (y₂-y₁)²]
  • Uses subtraction, squaring, and square root
  • Often results in irrational numbers

Both formulas work with coordinate pairs, but the midpoint finds a location while the distance finds a measurement.

Note: The midpoint is equidistant from both endpoints, which can be verified using the distance formula.