Solved Exercises on Graphing Geometric Shapes in Grade 7

Master graphing geometric shapes: plotting coordinates, drawing polygons, calculating perimeters, and understanding transformations through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Plotting Coordinates
Exercise 1
Plot the points A(3, 4), B(-2, 5), C(-4, -3), and D(1, -2) on the coordinate plane.
Definition:

Coordinate Plane: A two-dimensional plane formed by the intersection of a vertical y-axis and a horizontal x-axis.

Ordered Pair: A pair of numbers (x, y) that describes the location of a point on the coordinate plane.

Plotting Method:
  1. Start at the origin (0, 0)
  2. Move horizontally along the x-axis: positive to the right, negative to the left
  3. Move vertically along the y-axis: positive up, negative down
  4. Mark the point where the movements intersect
Point A
(3, 4)
Point B
(-2, 5)
Point C
(-4, -3)
Point D
(1, -2)
Step 1: Understanding Ordered Pairs

Each point has coordinates (x, y) where x is the horizontal distance and y is the vertical distance from the origin.

Step 2: Plotting Point A(3, 4)

Start at origin, move 3 units right (positive x), then 4 units up (positive y)

Step 3: Plotting Point B(-2, 5)

Start at origin, move 2 units left (negative x), then 5 units up (positive y)

Step 4: Plotting Point C(-4, -3)

Start at origin, move 4 units left (negative x), then 3 units down (negative y)

Step 5: Plotting Point D(1, -2)

Start at origin, move 1 unit right (positive x), then 2 units down (negative y)

Points plotted successfully on the coordinate plane
Final Answer:

Points A(3,4), B(-2,5), C(-4,-3), and D(1,-2) correctly plotted on the coordinate plane.

Applied Rules:

• Order Matters: Always plot x-coordinate first, then y-coordinate

• Signs Determine Direction: Positive moves right/up, negative moves left/down

• Quadrants: Points in different quadrants based on sign combinations

2 Drawing a Rectangle
Exercise 2
Draw a rectangle with vertices at A(2, 1), B(6, 1), C(6, 4), and D(2, 4). Find its perimeter and area.
Definition:

Rectangle: A quadrilateral with four right angles and opposite sides equal and parallel.

Vertices: The corner points of a geometric shape.

Vertices
A(2,1), B(6,1), C(6,4), D(2,4)
Length
|AB| = 4 units
Width
|AD| = 3 units
Step 1: Plot all vertices

Plot points A(2,1), B(6,1), C(6,4), and D(2,4) on the coordinate plane.

Step 2: Connect the vertices

Draw lines AB, BC, CD, and DA to form the rectangle.

Step 3: Calculate side lengths

Length (horizontal): |AB| = |6-2| = 4 units

Width (vertical): |AD| = |4-1| = 3 units

Step 4: Calculate perimeter and area

Perimeter = 2(length + width) = 2(4 + 3) = 14 units

Area = length × width = 4 × 3 = 12 square units

Perimeter = 14 units, Area = 12 square units
Final Answer:

Rectangle drawn with perimeter = 14 units and area = 12 square units.

Applied Rules:

• Distance Formula: Distance between (x₁,y₁) and (x₂,y₂) is |x₂-x₁| or |y₂-y₁| for horizontal/vertical lines

• Rectangle Properties: Opposite sides are equal and parallel

• Formulas: Perimeter = 2(l+w), Area = l×w

3 Drawing a Triangle
Exercise 3
Draw triangle ABC with vertices A(0, 0), B(4, 0), and C(2, 3). Find its area.
Definition:

Triangle: A polygon with three sides and three vertices.

Base and Height: Base is one side of the triangle, height is perpendicular distance from base to opposite vertex.

Vertices
A(0,0), B(4,0), C(2,3)
Base
|AB| = 4 units
Height
3 units
Step 1: Plot all vertices

Plot points A(0,0), B(4,0), and C(2,3) on the coordinate plane.

Step 2: Connect the vertices

Draw lines AB, BC, and CA to form the triangle.

Step 3: Identify base and height

Base AB = |4-0| = 4 units (horizontal line)

Height = perpendicular distance from C to AB = 3 units (vertical distance)

Step 4: Calculate area

Area = ½ × base × height = ½ × 4 × 3 = 6 square units

Area = 6 square units
Final Answer:

Triangle ABC drawn with area = 6 square units.

Applied Rules:

• Triangle Area: Area = ½ × base × height

• Perpendicular Height: Measure height perpendicular to the base

• Coordinate Geometry: Use coordinates to find distances and positions

Rules and methods, laws,...
\(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
Distance Formula
Area of Rectangle
\(A = l \times w\)
length × width
Perimeter of Rectangle
\(P = 2(l + w)\)
2 × (length + width)
Area of Triangle
\(A = \frac{1}{2}bh\)
½ × base × height
Key Definitions:

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

Ordered Pair: (x, y) representing a point's location

Vertex: A corner point of a geometric shape

Complete Methodology:
  1. Identify Coordinates: Note all given vertex coordinates
  2. Plot Points: Accurately place each point on the coordinate plane
  3. Connect Vertices: Draw straight lines between consecutive points
  4. Calculate Properties: Find distances, areas, perimeters as needed
  5. Verify Shape: Check if the figure meets the definition of the required shape
Tip 1: Always plot points in order and connect them sequentially to form the correct shape.
Tip 2: For rectangles, opposite sides will have the same length and be parallel.
Tip 3: When finding height of a triangle, measure perpendicular distance from base to opposite vertex.
Tip 4: Use the coordinate differences to easily calculate side lengths for horizontal and vertical lines.
Common Errors: Mixing up x and y coordinates, miscounting units, incorrect distance calculations.
Exam Preparation: Practice plotting various points, memorize area/perimeter formulas, understand coordinate relationships.
Solution: Exercises 4 to 5
4 Finding Side Lengths
Exercise 4
Given a parallelogram with vertices at P(1, 2), Q(5, 2), R(7, 5), and S(3, 5), find the lengths of all sides and determine if it's a rectangle.
Definition:

Parallelogram: A quadrilateral with opposite sides parallel and equal in length.

Rectangle: A parallelogram with four right angles.

Vertices
P(1,2), Q(5,2), R(7,5), S(3,5)
Side PQ
4 units
Side QR
√13 units
Step 1: Plot all vertices

Plot points P(1,2), Q(5,2), R(7,5), and S(3,5) on the coordinate plane.

Step 2: Calculate side lengths

PQ: Since y-coordinates are same, |PQ| = |5-1| = 4 units

QR: Using distance formula: √[(7-5)² + (5-2)²] = √[4+9] = √13 ≈ 3.61 units

RS: Since y-coordinates are same, |RS| = |7-3| = 4 units

SP: Using distance formula: √[(3-1)² + (5-2)²] = √[4+9] = √13 ≈ 3.61 units

Step 3: Check parallelogram properties

Opposite sides are equal: |PQ| = |RS| = 4 units and |QR| = |SP| = √13 units

This confirms it's a parallelogram.

Step 4: Check if it's a rectangle

For a rectangle, adjacent sides must be perpendicular.

Slope of PQ: (2-2)/(5-1) = 0 (horizontal line)

Slope of QR: (5-2)/(7-5) = 3/2

Since 0 × (3/2) ≠ -1, the sides are not perpendicular.

Therefore, it's a parallelogram but not a rectangle.

PQ = RS = 4 units, QR = SP = √13 units, Not a rectangle
Final Answer:

Sides: PQ = RS = 4 units, QR = SP = √13 units. It's a parallelogram but not a rectangle.

Applied Rules:

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

• Parallel Lines: Same slope, different y-intercepts

• Perpendicular Lines: Product of slopes = -1

5 Complex Polygon
Exercise 5
Draw a pentagon with vertices A(-2, 1), B(1, 3), C(3, 1), D(2, -2), and E(-1, -2). Find its approximate area using the coordinate geometry method.
Definition:

Pentagon: A five-sided polygon with five vertices.

Shoelace Formula: A method to calculate the area of a polygon given its vertices.

Vertices
A(-2,1), B(1,3), C(3,1), D(2,-2), E(-1,-2)
Shoelace
Formula
Area
≈ 13.5 units²
Step 1: Plot all vertices in order

Plot points A(-2,1), B(1,3), C(3,1), D(2,-2), and E(-1,-2) in sequence.

Step 2: Connect vertices to form pentagon

Draw lines AB, BC, CD, DE, and EA to complete the pentagon.

Step 3: Apply Shoelace Formula

Arrange coordinates in order and repeat first vertex at the end:

(-2,1), (1,3), (3,1), (2,-2), (-1,-2), (-2,1)

Sum of forward products: (-2×3) + (1×1) + (3×-2) + (2×-2) + (-1×1) = -6+1-6-4-1 = -16

Sum of backward products: (1×1) + (3×3) + (1×2) + (-2×-1) + (-2×-2) = 1+9+2+2+4 = 18

Area = ½|(-16) - 18| = ½|−34| = 17 square units

Step 4: Verify by decomposition

We can also divide the pentagon into triangles and rectangles for verification.

Area ≈ 17 square units
Final Answer:

Pentagon drawn with vertices A(-2,1), B(1,3), C(3,1), D(2,-2), E(-1,-2). Area = 17 square units.

Applied Rules:

• Shoelace Formula: Area = ½|x₁(y₂-yₙ) + x₂(y₃-y₁) + ... + xₙ(y₁-yₙ₋₁)|

• Vertex Order: List vertices in clockwise or counterclockwise order

• Complex Shapes: Break down into simpler shapes when possible

Key Concepts: Coordinate Geometry Fundamentals
\(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
Distance Formula
Key definitions:

Coordinate Plane: A two-dimensional surface defined by x and y axes intersecting at origin (0,0)

Ordered Pair: (x, y) representing a point's position with x as horizontal distance and y as vertical distance

Quadrants: Four regions created by axes: I(+,+), II(-,+), III(-,-), IV(+,-)

Complete methodology:
  1. Analyze the problem: Identify what shape needs to be drawn and what information is given
  2. Plot coordinates: Accurately place each vertex on the coordinate plane
  3. Connect vertices: Draw straight lines between consecutive points
  4. Calculate properties: Use coordinate differences or formulas to find lengths, areas, perimeters
  5. Verify results: Check if calculated values make sense geometrically
Tip 1: Always remember the order (x, y) - x comes first, then y.
Tip 2: For horizontal or vertical lines, distance is just the absolute difference of coordinates.
Tip 3: When plotting, go right/left first, then up/down.
Tip 4: For complex polygons, break them into simpler shapes to calculate area.
Common errors: Mixing up x and y coordinates, counting units incorrectly, forgetting negative signs.
Exam preparation: Practice plotting various points, memorize formulas, understand geometric properties.
Formulas to know by heart:

• Distance Formula: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)

• Midpoint Formula: \(M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\)

• Area of Rectangle: \(A = l \times w\)

• Perimeter of Rectangle: \(P = 2(l + w)\)

• Area of Triangle: \(A = \frac{1}{2}bh\)

Exercise with Visualization: Polygon Properties
Exercise 6: Analyzing Polygons
Consider the following polygons on the coordinate plane:
Square: A(0,0), B(4,0), C(4,4), D(0,4)
Rectangle: E(1,1), F(5,1), G(5,3), H(1,3)
Triangle: I(0,0), J(4,0), K(2,3)

Analysis: The visualization shows different polygon types and their coordinate representations.

  • Square: All sides equal, all angles 90°
  • Rectangle: Opposite sides equal, all angles 90°
  • Triangle: Three sides with varying properties

Questions & Answers

Question: I get confused about which number in the coordinate pair goes where. How do I remember which is x and which is y?

Answer: Great question! Here are helpful memory aids:

  • Alphabetical Order: In (x, y), 'x' comes before 'y' alphabetically, just like in the ordered pair.
  • Direction Mnemonic: "X comes first, go across" - x is horizontal (across), y is vertical.
  • Rhyme: "X is first, left and right; Y is second, up and down."

Think of it as reading: you go across the page first (x), then down to the next line (y). For example, in (3, 5), move 3 units right first, then 5 units up.

Always remember: (x-coordinate, y-coordinate) where x is horizontal movement and y is vertical movement.

Question: When I'm finding the area of a triangle on the coordinate plane, how do I know which side to use as the base?

Answer: You can choose any side as the base, but some choices make calculations easier:

  • Horizontal or Vertical Sides: If available, choose these as they make measuring height easier.
  • Longest Side: Often simplifies the calculation.
  • Any Side: The area will be the same regardless of which side you choose as the base.

Remember that the height must be measured perpendicular to the base. If your base is horizontal, the height is the vertical distance to the opposite vertex. If your base is vertical, the height is the horizontal distance.

For coordinates, if you have a horizontal base, the height is simply the difference in y-coordinates between the base and the opposite vertex.

Question: How do I know if a shape on the coordinate plane is a rectangle or just a parallelogram?

Answer: Here are the key differences and how to identify them:

  • Parallelogram: Opposite sides are parallel and equal in length.
  • Rectangle: A parallelogram with four right angles (90°).

To check if a parallelogram is a rectangle using coordinates:

  1. Verify opposite sides are equal (parallelogram property)
  2. Check if adjacent sides are perpendicular by calculating their slopes
  3. If the product of slopes of adjacent sides equals -1, they are perpendicular

Alternatively, you can check if the diagonals are equal in length - rectangles have equal diagonals while general parallelograms do not.

Question: When calculating distances between points that aren't horizontal or vertical, do I always need to use the distance formula?

Answer: Yes, for diagonal distances you must use the distance formula: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)

However, here are some shortcuts for specific situations:

  • Horizontal lines: d = |x₂ - x₁| (same y-coordinates)
  • Vertical lines: d = |y₂ - y₁| (same x-coordinates)
  • Special triangles: If the differences form a Pythagorean triple (3-4-5, 5-12-13, etc.), you can recognize the distance

The distance formula is derived from the Pythagorean theorem. Think of the line segment as the hypotenuse of a right triangle where the legs are the horizontal and vertical distances between the points.

Practice recognizing common coordinate differences that correspond to simple distances to speed up your calculations!

Question: How can I check if my plotted points and calculated measurements are correct?

Answer: Here are effective verification strategies:

  1. Visual Inspection: Does your shape look correct? Are sides roughly the expected length?
  2. Re-calculate: Perform the calculation a second time using a different method
  3. Estimation: Estimate the answer first to see if your calculation is reasonable
  4. Coordinate Checks: Verify that points satisfy the properties of the shape you're drawing

For example, if you're drawing a rectangle, check that opposite sides are equal and that all angles appear to be 90°. For triangles, ensure the sum of any two sides is greater than the third side.

Using graph paper helps with accuracy, and for more precise work, you can use online graphing tools to verify your manual calculations.