Solved Exercises on Midpoint Concept in Grade 7

Master midpoint calculations on the coordinate plane: midpoint formula, line segment applications, and geometric relationships through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Finding Midpoint of a Line Segment
Exercise 1
Find the midpoint of the line segment connecting points A(4, 6) and B(10, 2). Show your work using the midpoint formula.
Definition:

Midpoint: The point that divides a line segment into two equal parts. The midpoint formula for points (x₁, y₁) and (x₂, y₂) is: M = ((x₁+x₂)/2, (y₁+y₂)/2)

Midpoint calculation method:
  1. Identify coordinates: (x₁, y₁) and (x₂, y₂)
  2. Apply the midpoint formula to x-coordinates: (x₁+x₂)/2
  3. Apply the midpoint formula to y-coordinates: (y₁+y₂)/2
  4. Combine results to form the midpoint coordinates
Endpoints
A(4,6), B(10,2)
Midpoint Formula
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Calculated Midpoint
M(7, 4)
Step 1: Identify coordinates

A(4, 6): x₁ = 4, y₁ = 6

B(10, 2): x₂ = 10, y₂ = 2

Step 2: Calculate x-coordinate of midpoint

x_mid = (x₁ + x₂)/2 = (4 + 10)/2 = 14/2 = 7

Step 3: Calculate y-coordinate of midpoint

y_mid = (y₁ + y₂)/2 = (6 + 2)/2 = 8/2 = 4

Step 4: Combine coordinates

Midpoint M = (7, 4)

Midpoint = (7, 4)
Final answer:

The midpoint of the line segment connecting A(4, 6) and B(10, 2) is (7, 4)

Applied rules:

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

• Average calculation: Midpoint is the average of each coordinate

• Equal division: Midpoint divides the segment into two equal parts

2 Midpoint in Real-World Context
Exercise 2
Two cities are located at coordinates C₁(3, 7) and C₂(9, 1) on a map. A rest stop is to be built exactly halfway between the two cities. Where should the rest stop be located?
Definition:

Real-world midpoint: The midpoint formula applies to any situation where you need to find the middle point between two locations, whether physical or abstract.

City Locations
C₁(3,7), C₂(9,1)
Midpoint Formula
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Rest Stop Location
M(6, 4)
Step 1: Identify the coordinates

City C₁: (3, 7) and City C₂: (9, 1)

Step 2: Apply midpoint formula to x-coordinates

x_rest_stop = (3 + 9)/2 = 12/2 = 6

Step 3: Apply midpoint formula to y-coordinates

y_rest_stop = (7 + 1)/2 = 8/2 = 4

Step 4: State the answer in context

The rest stop should be located at coordinates (6, 4)

Rest stop location = (6, 4)
Final answer:

The rest stop should be built at coordinates (6, 4), exactly halfway between the two cities.

Applied rules:

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

• Real-world application: Midpoint represents the equal division point

• Problem-solving: Translate word problems into mathematical expressions

3 Midpoints of Triangle Sides
Exercise 3
Triangle ABC has vertices A(2, 4), B(8, 6), and C(6, 2). Find the coordinates of the midpoints of all three sides: AB, BC, and AC.
Definition:

Midsegment: A line segment connecting the midpoints of two sides of a triangle. The midpoint of each side is found using the midpoint formula.

Vertices
A(2,4), B(8,6), C(6,2)
Midpoints
M_AB(5,5), M_BC(7,4), M_AC(4,3)
Formula Applied
Three times
Step 1: Find midpoint of side AB

A(2, 4) and B(8, 6)

M_AB = ((2+8)/2, (4+6)/2) = (10/2, 10/2) = (5, 5)

Step 2: Find midpoint of side BC

B(8, 6) and C(6, 2)

M_BC = ((8+6)/2, (6+2)/2) = (14/2, 8/2) = (7, 4)

Step 3: Find midpoint of side AC

A(2, 4) and C(6, 2)

M_AC = ((2+6)/2, (4+2)/2) = (8/2, 6/2) = (4, 3)

Step 4: Verify the results

Each midpoint is exactly halfway between its respective endpoints

M_AB(5, 5), M_BC(7, 4), M_AC(4, 3)
Final answer:

The midpoints of the sides of triangle ABC are: M_AB(5, 5), M_BC(7, 4), and M_AC(4, 3)

Applied rules:

• Midpoint formula: Apply to each pair of vertices

• Triangle midsegments: Connect midpoints of sides

• Repetitive application: Same formula for multiple pairs

Rules and methods, laws,...
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Midpoint Formula
Midpoint Formula
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Coordinates of the midpoint
Average Concept
Midpoint = Average of coordinates
Understanding the principle
Equal Division
AM = MB when M is midpoint
Distance property
Key definitions:

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

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

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

Ordered pair: A set of two numbers (x, y) representing a point's location

Complete methodology:
  1. Identify endpoints: Determine the coordinates of both ends of the segment
  2. Apply formula to x-coordinates: Calculate (x₁+x₂)/2
  3. Apply formula to y-coordinates: Calculate (y₁+y₂)/2
  4. Combine results: Form the ordered pair for the midpoint
  5. Verify: Check that the midpoint is between the endpoints
Tip 1: Think of the midpoint as the "average" of the coordinates.
Tip 2: The midpoint is always between the two endpoints.
Tip 3: For horizontal segments, the y-coordinate stays the same.
Tip 4: For vertical segments, the x-coordinate stays the same.
Common errors: Adding coordinates instead of averaging, mixing up x and y values, calculation errors with fractions, forgetting to divide by 2.
Exam preparation: Practice with various coordinate combinations, memorize the midpoint formula, understand geometric implications, work on word problems involving midpoints.
Solution: Exercises 4 to 5
4 Finding Endpoint Given Midpoint
Exercise 4
The midpoint of a line segment is M(5, 3), and one endpoint is A(2, 7). Find the coordinates of the other endpoint B.
Definition:

Endpoint calculation: If M is the midpoint of segment AB, then M = ((x_A + x_B)/2, (y_A + y_B)/2). Solving for B: B = (2x_M - x_A, 2y_M - y_A).

Given Information
M(5,3), A(2,7)
Formula Manipulation
B = (2x_M - x_A, 2y_M - y_A)
Calculated Endpoint
B(8, -1)
Step 1: Write the midpoint formula

For midpoint M of segment AB: M = ((x_A + x_B)/2, (y_A + y_B)/2)

Step 2: Set up equations with known values

(5, 3) = ((2 + x_B)/2, (7 + y_B)/2)

Step 3: Solve for x-coordinate of B

5 = (2 + x_B)/2

10 = 2 + x_B

x_B = 8

Step 4: Solve for y-coordinate of B

3 = (7 + y_B)/2

6 = 7 + y_B

y_B = -1

Other endpoint B(8, -1)
Final answer:

The coordinates of the other endpoint B are (8, -1)

Applied rules:

• Midpoint formula rearrangement: Solve for unknown endpoint

• Algebraic manipulation: Isolate variables in equations

• Verification: Check that M is indeed the midpoint of A and B

5 Diagonal Midpoint of Rectangle
Exercise 5
Rectangle PQRS has vertices P(1, 1), Q(7, 1), R(7, 5), and S(1, 5). Find the midpoint of diagonal PR and verify that it's the same as the midpoint of diagonal QS.
Definition:

Rectangle diagonals: In a rectangle, the diagonals bisect each other, meaning they have the same midpoint.

Rectangle Vertices
P(1,1), Q(7,1), R(7,5), S(1,5)
Diagonal PR
P(1,1) to R(7,5)
Diagonal QS
Q(7,1) to S(1,5)
Step 1: Find midpoint of diagonal PR

P(1, 1) and R(7, 5)

M_PR = ((1+7)/2, (1+5)/2) = (8/2, 6/2) = (4, 3)

Step 2: Find midpoint of diagonal QS

Q(7, 1) and S(1, 5)

M_QS = ((7+1)/2, (1+5)/2) = (8/2, 6/2) = (4, 3)

Step 3: Compare the midpoints

M_PR = (4, 3) and M_QS = (4, 3)

They are identical!

Step 4: Conclusion

This confirms that the diagonals of a rectangle bisect each other

Both diagonals have midpoint (4, 3)
Final answer:

Both diagonals PR and QS have the same midpoint at (4, 3), verifying that rectangle diagonals bisect each other.

Applied rules:

• Midpoint formula: Apply to both diagonals

• Rectangle property: Diagonals bisect each other

• Geometric verification: Properties hold for all rectangles

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

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

Line segment: A portion of a line with two distinct endpoints

Coordinate geometry: The study of geometric figures using coordinate systems

Segment bisector: A point, line, or plane that divides a segment into two equal parts

Rectangle diagonals: In a rectangle, diagonals are equal in length and bisect each other

Complete methodology:
  1. Identify coordinates: Determine the exact coordinates of both endpoints
  2. Apply midpoint formula: Calculate (x₁+x₂)/2 and (y₁+y₂)/2
  3. Form ordered pair: Combine the results as (x_mid, y_mid)
  4. Verify result: Check that the midpoint lies between the endpoints
  5. Apply context: Use the result in the problem's context
Tip 1: Remember that midpoint is the average of coordinates: (sum of coordinates) ÷ 2.
Tip 2: The midpoint is always between the two endpoints on the coordinate plane.
Tip 3: For reverse problems (finding endpoint), use: B = (2x_M - x_A, 2y_M - y_A).
Tip 4: Always verify your answer by checking that the midpoint is equidistant from both endpoints.
Common errors: Forgetting to divide by 2, adding instead of averaging, coordinate mix-up, calculation errors with negative numbers.
Exam preparation: Master the midpoint formula, practice with various coordinate combinations, understand geometric implications, solve reverse problems (finding endpoints), recognize special properties of shapes.
Midpoint Rules:

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

• Reverse calculation: If M is midpoint of AB, then B = (2x_M - x_A, 2y_M - y_A)

• Rectangle diagonals: Bisect each other at the same point

• Equal distances: Distance from midpoint to each endpoint is equal

• Between endpoints: Midpoint coordinates are between the endpoint coordinates

Questions & Answers

Question: Why do we divide by 2 in the midpoint formula? What's the connection to averages?

Answer: Great question! The connection is very direct:

  • Average concept: The midpoint is literally the average of the x-coordinates and the average of the y-coordinates
  • Division by 2: When you find the average of two numbers, you add them together and divide by how many numbers there are (2)
  • Geometric meaning: The midpoint is exactly in the middle, so it's the "balanced" point between the two endpoints

Example: For points (4, 6) and (10, 2), the midpoint is the average of x-values (4+10)/2 = 7 and average of y-values (6+2)/2 = 4, giving us (7, 4).

Question: How do I find an endpoint if I know the midpoint and one endpoint? It seems like I need to work backwards.

Answer: Yes, you're working backwards! Here's the systematic approach:

  1. Start with the midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2)
  2. Solve for the unknown coordinate: If you know M and (x₁, y₁), then x₂ = 2x_M - x₁
  3. Apply to both coordinates: y₂ = 2y_M - y₁

Example: If midpoint M is (5, 3) and one endpoint A is (2, 7), then the other endpoint B is (2×5-2, 2×3-7) = (8, -1).

This works because the midpoint is the center point, so the other endpoint is the same distance from the midpoint as the known endpoint.

Question: Why do rectangle diagonals always have the same midpoint? Is this true for other shapes too?

Answer: This property holds for parallelograms, which includes rectangles:

  • Rectangle property: Opposite sides are equal and parallel, so diagonals bisect each other
  • Parallelogram property: Diagonals of any parallelogram bisect each other
  • Other shapes: This isn't true for triangles, trapezoids, or other quadrilaterals unless they're parallelograms

The reason is geometric: in a parallelogram, opposite sides are equal and parallel, which forces the diagonals to cross at their mutual midpoints.

This is a special property of parallelograms and is very useful for solving geometric problems!