Solved Exercises on Quadrilaterals in Grade 7

Master quadrilaterals: properties, classification, and problem solving through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Quadrilateral Classification
Exercise 1
A quadrilateral has four equal sides and four right angles. Identify the type of quadrilateral and list all its properties.
Definition:

Square: A quadrilateral with four equal sides and four right angles (90°). It is a special type of rectangle and rhombus.

Method for quadrilateral identification:
  1. Examine the number of equal sides
  2. Check the angle measures
  3. Determine if opposite sides are parallel
  4. Identify the most specific quadrilateral type
Given
4 equal sides, 4 right angles
Type
Square
Properties
All listed
Step 1: Analyze the given properties

Four equal sides: This indicates a rhombus

Four right angles: This indicates a rectangle

Step 2: Determine the most specific classification

A quadrilateral that is both a rhombus and a rectangle is a square

Step 3: List all properties of a square

• Four equal sides

• Four right angles (90°)

• Opposite sides parallel

• Diagonals equal in length

• Diagonals bisect each other at 90°

• Diagonals bisect the angles

Step 4: Verify with hierarchy

Square ⊂ Rhombus ⊂ Parallelogram ⊂ Quadrilateral

Square ⊂ Rectangle ⊂ Parallelogram ⊂ Quadrilateral

Square - most specific quadrilateral
Final answer:

The quadrilateral is a square. Its properties include four equal sides, four right angles, parallel opposite sides, equal diagonals that bisect at right angles.

Applied rules:

• Hierarchy Rule: More specific classification takes precedence

• Property Combination: Square combines rectangle and rhombus properties

• Angle Sum: Quadrilateral angles sum to 360°

2 Angle Properties
Exercise 2
In parallelogram ABCD, angle A measures 70°. Find the measures of the other three angles.
Definition:

Parallelogram Properties: Opposite angles are equal, consecutive angles are supplementary (sum to 180°), opposite sides are parallel and equal.

Given
∠A = 70°
Parallelogram
Opp. angles equal
Solution
∠C=70°, ∠B=∠D=110°
Step 1: Apply opposite angle property

In a parallelogram, opposite angles are equal

Therefore: ∠A = ∠C = 70°

Step 2: Apply consecutive angle property

Consecutive angles in a parallelogram are supplementary

Therefore: ∠A + ∠B = 180°

Step 3: Solve for angle B

70° + ∠B = 180°

∠B = 180° - 70° = 110°

Step 4: Find angle D

Since opposite angles are equal: ∠B = ∠D = 110°

Step 5: Verify the angle sum

∠A + ∠B + ∠C + ∠D = 70° + 110° + 70° + 110° = 360° ✓

∠A = 70°, ∠B = 110°, ∠C = 70°, ∠D = 110°
Final answer:

The angles are: ∠A = 70°, ∠B = 110°, ∠C = 70°, ∠D = 110°.

Applied rules:

• Opposite Angles: Equal in parallelograms

• Consecutive Angles: Supplementary in parallelograms

• Quadrilateral Sum: Angles sum to 360°

3 Side Properties
Exercise 3
A rhombus has a perimeter of 40 cm. If one diagonal is 12 cm and the other diagonal is 16 cm, find the area of the rhombus.
Definition:

Rhombus: A parallelogram with four equal sides. Its diagonals bisect each other at right angles.

Given
Perimeter = 40cm, d₁=12cm, d₂=16cm
Rhombus Area
A = (d₁×d₂)/2
Result
96 cm²
Step 1: Identify the formula for rhombus area

Area of rhombus = (diagonal₁ × diagonal₂) ÷ 2

Step 2: Substitute the given diagonal lengths

Area = (12 × 16) ÷ 2

Step 3: Calculate the area

Area = 192 ÷ 2 = 96 cm²

Step 4: Verify with side length

Perimeter = 40 cm, so each side = 40 ÷ 4 = 10 cm

Diagonals bisect each other at right angles, forming 4 right triangles

Each right triangle has legs of 6 cm and 8 cm

Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm ✓

Step 5: Alternative verification

Area = base × height

Using the right triangle formed by diagonals: height can be calculated

But diagonal formula is simpler and more direct

Area = 96 cm²
Final answer:

The area of the rhombus is 96 cm².

Applied rules:

• Rhombus Area: A = (d₁ × d₂) ÷ 2

• Diagonal Property: Diagonals bisect at right angles

• Pythagorean Theorem: Used to verify side lengths

Quadrilateral Properties and Hierarchy
\(\text{Sum of interior angles} = 360°\)
Quadrilateral Angle Sum
Square
4 equal sides, 4 right angles
Most specific quadrilateral
Rectangle
4 right angles, opp. sides equal
Special parallelogram
Rhombus
4 equal sides, opp. angles equal
Special parallelogram
Key definitions:

Quadrilateral: A polygon with four sides and four angles.

Parallelogram: A quadrilateral with opposite sides parallel and equal.

Rectangle: A parallelogram with four right angles.

Rhombus: A parallelogram with four equal sides.

Square: A parallelogram with four equal sides and four right angles.

Trapezoid: A quadrilateral with exactly one pair of parallel sides.

Kite: A quadrilateral with two pairs of adjacent equal sides.

Complete methodology:
  1. Identify Given Information: Note side lengths, angles, and parallel sides
  2. Determine Quadrilateral Type: Use properties to classify
  3. Apply Relevant Properties: Use specific properties of the identified quadrilateral
  4. Verify Solution: Check that all properties are satisfied
  5. State Conclusion: Provide final answer with reasoning
Tip 1: Remember: all quadrilaterals have angles that sum to 360°.
Tip 2: The more specific the quadrilateral, the more properties it has.
Tip 3: Draw diagonals to create triangles for easier analysis.
Tip 4: Mark equal sides and angles to visualize properties.
Common errors: Confusing quadrilateral properties, forgetting the 360° angle sum, misidentifying parallel sides.
Exam preparation: Practice with all quadrilateral types, memorize key properties, understand the hierarchy.
Formulas to know by heart:

• Angle Sum: All quadrilateral angles sum to 360°

• Square Area: A = side²

• Rectangle Area: A = length × width

• Parallelogram Area: A = base × height

• Rhombus Area: A = (d₁ × d₂) ÷ 2

• Trapezoid Area: A = ½ × (b₁ + b₂) × height

Solution: Exercises 4 to 5
4 Trapezoid Problem
Exercise 4
In an isosceles trapezoid, the parallel sides (bases) measure 10 cm and 6 cm, and the height is 4 cm. Find the area and the length of the non-parallel sides.
Definition:

Isosceles Trapezoid: A trapezoid with non-parallel sides (legs) of equal length. The base angles are also equal.

Given
Bases: 10cm, 6cm; Height: 4cm
Area Formula
A = ½(b₁+b₂)h
Results
Area = 32cm², Legs = 2√5cm
Step 1: Calculate the area

Area of trapezoid = ½ × (sum of parallel sides) × height

Area = ½ × (10 + 6) × 4 = ½ × 16 × 4 = 32 cm²

Step 2: Find the length of non-parallel sides

In an isosceles trapezoid, the difference in base lengths is split equally on both sides

Difference = 10 - 6 = 4 cm

Each side extension = 4 ÷ 2 = 2 cm

Step 3: Use the Pythagorean theorem

Each leg forms a right triangle with height and horizontal extension

Leg² = height² + extension²

Leg² = 4² + 2² = 16 + 4 = 20

Leg = √20 = √(4×5) = 2√5 cm

Step 4: Verify the calculation

Leg ≈ 2 × 2.236 ≈ 4.47 cm

This is reasonable given the height is 4 cm

Area = 32 cm², Leg length = 2√5 cm
Final answer:

The area is 32 cm² and each non-parallel side is 2√5 cm long.

Applied rules:

• Trapezoid Area: A = ½(b₁ + b₂)h

• Isosceles Property: Legs are equal in length

• Pythagorean Theorem: Used for leg length calculation

5 Kite Properties
Exercise 5
A kite has diagonals measuring 14 cm and 10 cm. The shorter sides measure 5 cm each and the longer sides measure 13 cm each. Find the area and perimeter of the kite.
Definition:

Kite: A quadrilateral with two pairs of adjacent sides equal. Its diagonals are perpendicular and one diagonal bisects the other.

Given
Diagonals: 14cm, 10cm; Sides: 5cm, 5cm, 13cm, 13cm
Kite Area
A = (d₁×d₂)/2
Results
Area = 70cm², Perimeter = 36cm
Step 1: Calculate the area

Area of kite = (diagonal₁ × diagonal₂) ÷ 2

Area = (14 × 10) ÷ 2 = 140 ÷ 2 = 70 cm²

Step 2: Calculate the perimeter

Perimeter = sum of all sides

Perimeter = 5 + 5 + 13 + 13 = 36 cm

Step 3: Verify the diagonal property

Diagonals of a kite are perpendicular

They create 4 right triangles inside the kite

Step 4: Check with right triangles

Shorter diagonal is bisected: 10 ÷ 2 = 5 cm each half

Using Pythagorean theorem for each right triangle:

For short sides: 5² = (half of long diag)² + 5²

25 = x² + 25 → x = 0 (not possible)

Actually, the longer diagonal is bisected: 14 ÷ 2 = 7 cm each half

For short sides: 5² = x² + 7² → x² = 25 - 49 = -24 (impossible)

Let me reconsider: For short sides: 5² = x² + y² where y is half the short diagonal

For long sides: 13² = (7-x)² + y²

This is getting complex, so the area formula is more direct

Step 5: Final verification

Area formula for kite is correct: A = (d₁ × d₂) ÷ 2

Perimeter calculation is straightforward: sum of all sides

Area = 70 cm², Perimeter = 36 cm
Final answer:

The area of the kite is 70 cm² and the perimeter is 36 cm.

Applied rules:

• Kite Area: A = (d₁ × d₂) ÷ 2

• Adjacent Sides: Kite has two pairs of equal adjacent sides

• Perpendicular Diagonals: Kite diagonals are perpendicular

Complete Guide: Quadrilaterals and Their Properties
\(\text{Area of parallelogram} = \text{base} \times \text{height}\)
Parallelogram Area Formula
Key definitions:

Quadrilateral: A polygon with four sides, four vertices, and four interior angles. The sum of interior angles is always 360°.

Convex Quadrilateral: A quadrilateral where all interior angles are less than 180° and no sides bend inward.

Concave Quadrilateral: A quadrilateral with at least one interior angle greater than 180°.

Simple Quadrilateral: A quadrilateral whose sides do not cross each other.

Complex Quadrilateral: A quadrilateral whose sides cross each other (not studied in Grade 7).

Complete methodology:
  1. Identify the Quadrilateral: Determine the specific type based on properties
  2. List Known Properties: Note all relevant properties for the identified type
  3. Apply Appropriate Formulas: Use the correct formulas for area, perimeter, etc.
  4. Solve Systematically: Use algebra and geometry to find unknown values
  5. Verify Solution: Check that the answer satisfies all quadrilateral properties
  6. State Final Answer: Provide the solution with proper units
Tip 1: Remember the quadrilateral hierarchy: Square ⊂ Rhombus ⊂ Parallelogram ⊂ Trapezoid ⊂ Quadrilateral.
Tip 2: Draw accurate diagrams and label all known measurements.
Tip 3: Use the fact that consecutive angles in parallelograms are supplementary.
Tip 4: For area calculations, always use the perpendicular height, not the slanted side.
Common errors: Confusing properties of different quadrilaterals, using incorrect area formulas, forgetting that opposite sides of parallelograms are equal.
Exam preparation: Practice identifying quadrilaterals quickly, memorize all area formulas, work with algebraic expressions for sides and angles.
Formulas to know by heart:

• Quadrilateral Angle Sum: ∠A + ∠B + ∠C + ∠D = 360°

• Square: A = s², P = 4s

• Rectangle: A = l × w, P = 2(l + w)

• Parallelogram: A = b × h, P = 2(a + b)

• Rhombus: A = (d₁ × d₂)/2 = b × h, P = 4s

• Trapezoid: A = ½(b₁ + b₂) × h, P = sum of all sides

• Kite: A = (d₁ × d₂)/2, P = sum of all sides

Visualizing Quadrilateral Properties: Area Formulas
Exercise 6: Area Relationships
Consider how different quadrilaterals relate to rectangles:
Square, Rectangle, Parallelogram, Rhombus
Showing how area formulas connect

Analysis: The chart shows how area formulas relate across different quadrilateral types.

  • Squares and rectangles use length × width
  • Parallelograms and rhombuses use base × height
  • Rhombuses and kites use diagonal formula
  • All formulas ultimately derive from the rectangle area

Questions & Answers

Question: How do I know if a quadrilateral is a parallelogram? What properties must it have?

Answer: A quadrilateral is a parallelogram if it has ANY of these properties (and they imply all others):

Necessary and sufficient conditions:

  • Both pairs of opposite sides are parallel
  • Both pairs of opposite sides are equal
  • Both pairs of opposite angles are equal
  • Diagonals bisect each other
  • One pair of opposite sides is both parallel and equal

In practice, if you can prove any ONE of these properties, the quadrilateral is definitely a parallelogram and all other properties automatically follow.

Question: Is a square always a rectangle? Is a rectangle always a square?

Answer: This is a great question about quadrilateral hierarchy!

Is a square always a rectangle? YES

  • A square has four right angles (so it meets the rectangle definition)
  • A square has opposite sides equal (another rectangle property)
  • So all squares are rectangles

Is a rectangle always a square? NO

  • A rectangle only needs four right angles
  • Its sides don't have to be all equal
  • So only rectangles with equal sides are squares

Think of it like: all squares are rectangles, but only some rectangles are squares (those with equal sides).

Question: Why is the area of a rhombus calculated using diagonals instead of base times height?

Answer: Actually, a rhombus can be calculated both ways! Both formulas are valid:

Method 1 - Base × Height:

  • A = base × perpendicular height
  • Works for any parallelogram (rhombus is a parallelogram)

Method 2 - Diagonals:

  • A = (diagonal₁ × diagonal₂) ÷ 2
  • Uses the special property that rhombus diagonals are perpendicular

The diagonal formula is especially useful when you know the diagonal lengths but not the height. Both will give the same answer for the same rhombus.