Solved Exercises on Geometric Figures in Coordinate Planes in Grade 8

Master geometric figures: plotting points, calculating area and perimeter, transformations, and properties through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Rectangle on Coordinate Plane
Exercise 1
Plot the rectangle with vertices A(2, 1), B(2, 5), C(6, 5), and D(6, 1). Find its area and perimeter.
Definition:

Rectangle on Coordinate Plane: A four-sided polygon with four right angles. Opposite sides are equal and parallel. In coordinate geometry, we can calculate area and perimeter using coordinate values.

Rectangle Calculation Method:
  1. Plot all vertices on the coordinate plane
  2. Connect consecutive vertices to form the rectangle
  3. Calculate length and width using coordinate differences
  4. Area = length × width
  5. Perimeter = 2(length + width)
Vertices
A(2,1), B(2,5), C(6,5), D(6,1)
Length & Width
l = 4, w = 4
Area & Perimeter
A = 16, P = 16
Step 1: Identify coordinates

A(2, 1), B(2, 5), C(6, 5), D(6, 1)

Step 2: Calculate side lengths

Length (AB): |5 - 1| = 4 units (vertical side)

Width (AD): |6 - 2| = 4 units (horizontal side)

Step 3: Calculate area

Area = length × width = 4 × 4 = 16 square units

Step 4: Calculate perimeter

Perimeter = 2(length + width) = 2(4 + 4) = 2(8) = 16 units

Area = 16 sq units, Perimeter = 16 units
Final answer:

The rectangle has an area of 16 square units and a perimeter of 16 units.

Applied rules:

• Rectangle Properties: Opposite sides equal and parallel

• Coordinate Distance: Distance between points with same x/y coordinate is absolute difference

• Area Formula: A = length × width

2 Triangle Area Calculation
Exercise 2
Find the area of triangle PQR with vertices P(1, 2), Q(5, 2), and R(3, 6). Use the coordinate formula for area.
Definition:

Triangle Area Formula: For a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃), the area is: A = (1/2)|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|.

Given Vertices
P(1,2), Q(5,2), R(3,6)
Formula Substitution
A = (1/2)|1(2-6) + 5(6-2) + 3(2-2)|
Result
A = 8 sq units
Step 1: Assign coordinates

(x₁, y₁) = (1, 2), (x₂, y₂) = (5, 2), (x₃, y₃) = (3, 6)

Step 2: Apply the formula

A = (1/2)|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|

A = (1/2)|1(2 - 6) + 5(6 - 2) + 3(2 - 2)|

Step 3: Calculate each term

Term 1: 1(2 - 6) = 1(-4) = -4

Term 2: 5(6 - 2) = 5(4) = 20

Term 3: 3(2 - 2) = 3(0) = 0

Step 4: Sum and find absolute value

A = (1/2)|-4 + 20 + 0| = (1/2)|16| = (1/2)(16) = 8

A = 8 square units
Final answer:

The area of triangle PQR is 8 square units.

Applied rules:

• Shoelace Formula: A = (1/2)|∑(xᵢyᵢ₊₁ - xᵢ₊₁yᵢ)|

• Absolute Value: Area is always positive

• Order Matters: Follow vertices in consistent order (clockwise or counterclockwise)

3 Parallelogram Properties
Exercise 3
Verify that the points A(1, 1), B(4, 1), C(6, 4), and D(3, 4) form a parallelogram by showing that opposite sides are equal and parallel.
Definition:

Parallelogram: A quadrilateral with both pairs of opposite sides parallel and equal in length. Diagonals bisect each other.

Given Points
A(1,1), B(4,1), C(6,4), D(3,4)
Opposite Sides
AB = DC = 3, AD = BC = 3√2
Conclusion
It's a parallelogram
Step 1: Calculate length of side AB

AB = √[(4-1)² + (1-1)²] = √[9 + 0] = √9 = 3

Step 2: Calculate length of side DC

DC = √[(6-3)² + (4-4)²] = √[9 + 0] = √9 = 3

Step 3: Calculate length of side AD

AD = √[(3-1)² + (4-1)²] = √[4 + 9] = √13

Step 4: Calculate length of side BC

BC = √[(6-4)² + (4-1)²] = √[4 + 9] = √13

Step 5: Verify opposite sides are equal

AB = DC = 3 and AD = BC = √13

Since opposite sides are equal, ABCD is a parallelogram.

ABCD is a parallelogram
Final answer:

Since opposite sides are equal (AB = DC = 3 and AD = BC = √13), the points form a parallelogram.

Applied rules:

• Distance Formula: d = √[(x₂-x₁)² + (y₂-y₁)²]

• Parallelogram Property: Opposite sides are equal and parallel

• Verification: Measure all four sides to confirm equality of opposites

Rules and methods, laws,...
\(A = \frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|\)
Triangle Area Formula
Rectangle Area
A = l × w
Length times width
Rectangle Perimeter
P = 2(l + w)
Twice the sum of length and width
Triangle Area
A = (1/2)bh
Half base times height
Distance Formula
d = √[(x₂-x₁)² + (y₂-y₁)²]
Distance between two points
Midpoint Formula
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Point at center of segment
Coordinate Geometry: Uses algebraic methods to study geometric figures
Properties Preserved: Geometric properties remain the same when figures are placed on coordinate planes
Key definitions:

Coordinate Geometry: The branch of mathematics that studies geometric figures using coordinate systems.

Vertex: A corner point of a geometric figure where two or more lines meet.

Diagonal: A line segment connecting two non-adjacent vertices of a polygon.

Perimeter: The total distance around the boundary of a geometric figure.

Area: The amount of space inside a two-dimensional figure, measured in square units.

Complete methodology:
  1. Plot vertices: Carefully place all points on the coordinate plane
  2. Connect points: Draw line segments in the correct order
  3. Identify properties: Determine if the figure is a rectangle, triangle, etc.
  4. Calculate measurements: Use appropriate formulas for area and perimeter
  5. Verify properties: Check if the figure has expected characteristics
  6. Express results: Include proper units
Tip 1: Always plot points carefully to avoid calculation errors.
Tip 2: For rectangles/squares aligned with axes, use coordinate differences directly.
Tip 3: Remember that area is always expressed in square units.

Solution: Exercises 4 to 5
4 Trapezoid on Coordinate Plane
Exercise 4
Find the area of trapezoid with vertices E(1, 1), F(7, 1), G(5, 4), and H(3, 4). Verify that it has one pair of parallel sides.
Definition:

Trapezoid: A quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases, and the perpendicular distance between them is the height.

Given Vertices
E(1,1), F(7,1), G(5,4), H(3,4)
Parallel Sides
EF || HG
Area
A = 12 sq units
Step 1: Verify parallel sides

Slope of EF = (1-1)/(7-1) = 0/6 = 0

Slope of HG = (4-4)/(5-3) = 0/2 = 0

Since both slopes are 0, EF || HG

Step 2: Calculate lengths of parallel sides

Base₁ (EF) = |7-1| = 6 units

Base₂ (HG) = |5-3| = 2 units

Step 3: Find the height

Height = |4-1| = 3 units (difference in y-coordinates)

Step 4: Calculate area

Area = (1/2)(base₁ + base₂) × height

Area = (1/2)(6 + 2) × 3 = (1/2)(8) × 3 = 12 square units

A = 12 square units
Final answer:

The area of the trapezoid is 12 square units, with parallel sides EF and HG.

Applied rules:

• Slope Test: Parallel lines have equal slopes

• Trapezoid Area: A = (1/2)(b₁ + b₂)h

• Coordinate Distance: For horizontal lines, distance is absolute difference of x-coordinates

5 Polygon Transformation
Exercise 5
Triangle ABC has vertices A(2, 3), B(4, 1), and C(6, 5). Translate the triangle 3 units left and 2 units up. Find the new coordinates and verify that the area remains unchanged.
Definition:

Translation: A transformation that moves every point of a figure the same distance in the same direction. The rule is (x, y) → (x+h, y+k) where h and k are the horizontal and vertical shifts.

Original Triangle
A(2,3), B(4,1), C(6,5)
Translation Rule
(x,y) → (x-3, y+2)
New Coordinates
A'(-1,5), B'(1,3), C'(3,7)
Step 1: Apply translation rule (x, y) → (x-3, y+2)

A(2, 3) → A'(2-3, 3+2) = A'(-1, 5)

B(4, 1) → B'(4-3, 1+2) = B'(1, 3)

C(6, 5) → C'(6-3, 5+2) = C'(3, 7)

Step 2: Calculate original area

A = (1/2)|2(1-5) + 4(5-3) + 6(3-1)|

A = (1/2)|2(-4) + 4(2) + 6(2)| = (1/2)|-8 + 8 + 12| = (1/2)(12) = 6

Step 3: Calculate translated area

A' = (1/2)|(-1)(3-7) + 1(7-5) + 3(5-3)|

A' = (1/2)|(-1)(-4) + 1(2) + 3(2)| = (1/2)|4 + 2 + 6| = (1/2)(12) = 6

Step 4: Verify

Original area = 6 sq units, Translated area = 6 sq units

Area is preserved under translation!

A'(-1,5), B'(1,3), C'(3,7), Area = 6 sq units
Final answer:

The translated triangle has vertices A'(-1, 5), B'(1, 3), C'(3, 7) with the same area of 6 square units.

Applied rules:

• Translation Rule: (x, y) → (x+h, y+k)

• Isometry: Translations preserve distances and angles

• Area Conservation: Area remains unchanged under rigid transformations

Geometric Figures Laws, Methods, and Properties
\(A = \frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|\)
Shoelace Formula
Key definitions:

Coordinate Geometry: The study of geometric figures using coordinate systems to analyze properties and relationships.

Rigid Transformation: A transformation that preserves distances and angles. Includes translations, rotations, and reflections.

Similarity Transformation: A transformation that preserves shape but not necessarily size. Includes dilations.

Complete methodology:
  1. Identify vertices: List all coordinates of the geometric figure
  2. Determine figure type: Classify based on properties (number of sides, angles, etc.)
  3. Calculate measurements: Use appropriate formulas for area, perimeter, etc.
  4. Verify properties: Check if the figure meets the criteria for its classification
  5. Apply transformations: Use coordinate rules for translations, rotations, reflections
  6. Preserve invariants: Note which properties remain unchanged
Tip 1: Always verify that the order of vertices is consistent when applying area formulas.
Tip 2: Rigid transformations preserve area, perimeter, and angle measures.

Tip 3: Use the slope formula to verify parallelism and perpendicularity.

Common errors: Misordering vertices in area formulas, forgetting to take absolute value, confusing area and perimeter calculations, incorrectly identifying parallel sides.
Exam preparation: Practice plotting points accurately, memorize area formulas, understand transformation rules, verify geometric properties using coordinates.

Coordinate geometry properties:

• Rectangle: Opposite sides equal and parallel, all angles 90°

• Parallelogram: Opposite sides equal and parallel, opposite angles equal

• Rhombus: All sides equal, diagonals perpendicular

• Square: All sides equal, all angles 90°, diagonals equal and perpendicular

• Triangle Area: A = (1/2)bh or using coordinate formula

• Trapezoid Area: A = (1/2)(b₁ + b₂)h

• Distance Preservation: Rigid transformations maintain all distances and angles

Exercise with Visualization: Polygon Properties
Exercise 6: Area vs Perimeter Relationships
Consider different geometric figures with the same perimeter:
Square: side length 4, perimeter 16, area 16
Rectangle: 3×5, perimeter 16, area 15
Triangle: base 6, height 4, perimeter ≈ 15.2, area 12

Analysis: The chart shows how area varies with different shapes having similar perimeters.

  • Squares generally have maximum area for a given perimeter
  • More compact shapes tend to have larger areas
  • Shape affects the area-perimeter relationship

Questions & Answers

Question: How do I know if a quadrilateral is a parallelogram, rectangle, or square using coordinates?

Answer: Check in this order:

  • Parallelogram: Opposite sides equal and parallel (check both distance and slope)
  • Rectangle: Parallelogram with one right angle (check if adjacent sides are perpendicular)
  • Rhombus: Parallelogram with all sides equal
  • Square: Rectangle AND rhombus (all sides equal, all angles 90°)

Use the distance formula to check equal lengths and the slope formula to check parallelism (equal slopes) and perpendicularity (negative reciprocal slopes).

Question: Why does the shoelace formula work for finding the area of a polygon?

Answer: The shoelace formula works by decomposing the polygon into triangles and calculating their signed areas. The "cross-products" in the formula account for the area contributions of each triangular section. The name comes from the crisscross pattern of multiplication, similar to lacing a shoe.

For a triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃), it's essentially calculating the area using the cross product of vectors.

Question: What happens to the area and perimeter when I rotate a figure?

Answer: Rotation is a rigid transformation, which means it preserves distances and angles. Therefore:

  • Area remains unchanged
  • Perimeter remains unchanged
  • Side lengths remain unchanged
  • Angle measures remain unchanged

Only the position and orientation of the figure change during rotation!

Question: Can I use the coordinate plane to find the area of irregular polygons?

Answer: Yes! For any polygon with known vertex coordinates, you can use the shoelace formula:

A = (1/2)|∑(xᵢyᵢ₊₁ - xᵢ₊₁yᵢ)|

This works for any simple polygon (one that doesn't intersect itself). List the vertices in order (clockwise or counterclockwise), and apply the formula.

Question: How do I find the center of a polygon using coordinates?

Answer: For the centroid (geometric center) of a polygon with vertices (x₁,y₁), (x₂,y₂), ..., (xₙ,yₙ), the coordinates are:

Centroid x = (x₁ + x₂ + ... + xₙ)/n

Centroid y = (y₁ + y₂ + ... + yₙ)/n

This is simply the average of all x-coordinates and the average of all y-coordinates!