Quadrilaterals
Key Concepts
Solved Examples: 45
Practice Questions: 5
Solved Examples
Question 1: The interior angles of a quadrilateral are in the ratio 2 : 3 : 4 : 9. Find the measure of each angle.
2x + 3x + 4x + 9x = 360°
18x = 360°
x = 20°
Therefore,
2x = 40°
3x = 60°
4x = 80°
9x = 180°
The angles are 40°, 60°, 80° and 180°.
Question 2: The four interior angles of a quadrilateral are consecutive multiples of 15°. Find the angles.
15x, 15(x + 1), 15(x + 2) and 15(x + 3).
15x + 15(x + 1) + 15(x + 2) + 15(x + 3) = 360°
60x + 90 = 360°
60x = 270°
x = 4.5
Therefore, the angles are:
67.5°, 82.5°, 97.5° and 112.5°
Question 3: The angles of a quadrilateral are x°, (x + 20)°, (2x − 10)° and (3x − 30)°. Find the value of x.
7x − 20 = 360°
7x = 380°
x = 380⁄7
Question 4: Two opposite angles of a quadrilateral are equal. The other two angles are 65° and 115°. Find the equal angles.
x + x + 65° + 115° = 360°
2x + 180° = 360°
2x = 180°
x = 90°
Question 5: One angle of a quadrilateral is equal to the sum of the other three angles. Find the largest angle.
Then,
x = 360° − x
2x = 360°
x = 180°
Question 6: The four interior angles of a quadrilateral are consecutive even numbers. Find the angles.
x°, (x + 2)°, (x + 4)° and (x + 6)°.
x + (x + 2) + (x + 4) + (x + 6) = 360°
4x + 12 = 360°
4x = 348°
x = 87°
Therefore, the angles are:
87°, 89°, 91° and 93°
Question 7: Three angles of a quadrilateral are in the ratio 2 : 3 : 5. The fourth angle is 120°. Find the remaining three angles.
2x, 3x and 5x.
2x + 3x + 5x + 120° = 360°
10x = 240°
x = 24°
Therefore,
2x = 48°
3x = 72°
5x = 120°
The remaining three angles are 48°, 72° and 120°.
Question 8: A quadrilateral has all four sides equal and all four angles equal. Identify the quadrilateral.
Question 9: A quadrilateral has opposite sides equal and parallel. Its diagonals are equal but do not intersect at right angles. Identify the quadrilateral.
- Opposite sides equal
- Opposite sides parallel
- Diagonals equal
- Diagonals are not perpendicular
These are the properties of a rectangle.
Question 10: A quadrilateral has all four sides equal. Its diagonals bisect each other at right angles but are not equal. Identify the quadrilateral.
- All sides equal
- Diagonals bisect each other
- Diagonals are perpendicular
- Diagonals are not equal
These are the properties of a rhombus.
Question 11: A quadrilateral has exactly one pair of opposite sides parallel. The non-parallel sides are equal and the diagonals are equal. Identify the quadrilateral.
- One pair of opposite sides parallel
- Non-parallel sides equal
- Diagonals equal
These are the properties of an isosceles trapezium.
Question 12: A quadrilateral has opposite angles equal and adjacent angles supplementary. One diagonal bisects the other, but the diagonals are not equal. Identify the quadrilateral.
- Opposite angles equal
- Adjacent angles supplementary
- One diagonal bisects the other (equivalently, the diagonals bisect each other)
- Diagonals are not equal
These are the properties of a parallelogram.
Question 13: For what value of Y is the seven-digit number 46393Y8 divisible by 11?
Sum of digits in odd positions: = 4 + 3 + 3 + 8 = 18
Sum of digits in even positions:
= 6 + 9 + Y
= 15 + Y
Now, because: The maximum possible difference here is between 18 and 24 (since Y is at most 9), the difference can only lie between −6 and 3.
It can never be ±11.
Therefore, the only possible value is 0.
Therefore, 18 − (15 + Y) = 0
3 − Y = 0
∴ Y = 3
Question 14: In parallelogram ABCD, ∠A = (3x + 20)° and ∠B = (5x − 12)°. Find the value of x and all four angles of the parallelogram.
Therefore,
(3x + 20)° + (5x − 12)° = 180°
8x + 8 = 180
8x = 172
x = 21.5
∠A = (3 × 21.5 + 20)° = 84.5°
∠B = (5 × 21.5 − 12)° = 95.5°
Opposite angles are equal.
∠C = 84.5°
∠D = 95.5°
So, x = 21.5; ∠A = ∠C = 84.5°, ∠B = ∠D = 95.5°
Question 15: The diagonals of a parallelogram intersect at O. If AC = (5x + 6) cm and AO = (2x + 9) cm, find the value of x and the length of AC.
Therefore,
AO = AC ÷ 2
2(2x + 9) = 5x + 6
4x + 18 = 5x + 6
x = 12
AC = 5(12) + 6 = 66 cm
So, x = 12, AC = 66 cm
Question 16: The perimeter of a parallelogram is 124 cm. One side is 11 cm longer than the adjacent side. Find the lengths of the sides.
Let the shorter side be x cm.
Longer side = (x + 11) cm
Perimeter = 2(x + x + 11)
124 = 4x + 22
4x = 102
x = 25.5
Longer side = 36.5 cm
The sides are 25.5 cm and 36.5 cm
Question 17: In a parallelogram, one angle is twice its adjacent angle. Find all the interior angles.
Adjacent angle = 2x°.
Adjacent angles are supplementary.
x + 2x = 180°
3x = 180°
x = 60°
Other angle = 120°
Opposite angles are equal.
The angles are 60°, 120°, 60° and 120°.
Question 18: In parallelogram ABCD, the diagonals intersect at O. If AO : BO = 3 : 2 and AO = 18 cm, find AC and BD.
Since diagonals bisect each other,
AC = 36 cm
AO : BO = 3 : 2
BO = (2/3) × 18
= 12 cm
Therefore,
BD = 24 cm
Question 19: In a parallelogram, one angle is 30° less than twice its adjacent angle. Find all the interior angles.
Adjacent angle = 180° − x°.
According to the question,
x = 2(180° − x) − 30°
x = 360° − 2x − 30°
3x = 330°
x = 110°
Adjacent angle = 70°
Opposite angles are equal.
The angles are 110°, 70°, 110° and 70°.
Question 20: In rectangle ABCD, diagonal AC = (5x + 4) cm and diagonal BD = (7x − 12) cm. Find the value of x and the length of each diagonal.
Therefore,
5x + 4 = 7x − 12
2x = 16
x = 8
Hence,
AC = 5 × 8 + 4 = 44 cm
BD = 7 × 8 − 12 = 44 cm
So, x = 8, AC = BD = 44 cm
Question 21: The diagonals of a rectangle intersect at O. If AO = (3x + 7) cm and AC = (8x − 2) cm, find the value of x and the length of diagonal AC.
Therefore,
AO = AC ÷ 2
2(3x + 7) = 8x − 2
6x + 14 = 8x − 2
2x = 16
x = 8
AC = 8 × 8 − 2
= 62 cm
So, x = 8, AC = 62 cm
Question 22: The perimeter of a rectangle is 136 cm. The length is 16 cm more than the breadth. Find the length of each diagonal.
Length = (x + 16) cm
2(x + x + 16) = 136
4x + 32 = 136
x = 26
Length = 42 cm
Using Pythagoras’ theorem,
Diagonal² = 42² + 26²
= 1764 + 676
= 2440
Diagonal = √2440
Question 23: The diagonal of a rectangle is 25 cm and one side is 7 cm shorter than the other. Find the dimensions of the rectangle.
Shorter side = (x − 7) cm.
Using Pythagoras’ theorem,
x² + (x − 7)² = 25²
2x² − 14x − 576 = 0
x² − 7x − 288 = 0
(x − 18)(x + 16) = 0
x = 18
Other side = 11
The dimensions are 18 cm × 11 cm
Question 24: A rectangle and a square have the same perimeter of 96 cm. The length of the rectangle is 8 cm more than its breadth. Find the difference between their areas.
Let breadth = x cm
Length = x + 8
2(2x + 8) = 96
2x + 8 = 48
x = 20
Length = 28
Area = 560 cm²
Square
Side = 96 ÷ 4 = 24 cm
Area = 576 cm²
Difference
= 576 − 560
= 16 cm²
Question 25: A rectangle has the same diagonal as a square whose side is 20 cm. If one side of the rectangle is 16 cm, find the other side.
= √(20² + 20²)
= 20√2 cm
Let the unknown side of the rectangle be x cm.
Using Pythagoras’ theorem,
16² + x² = (20√2)²
256 + x² = 800
x² = 544
x = 4√34 cm
Question 26: The diagonal of a square is 10 cm longer than its side. Find the side of the square.
Diagonal = x√2
Given,
x√2 = x + 10
x(√2 − 1) = 10
x = 10/(√2 − 1)
Rationalising,
x = 10(√2 + 1)
Question 27: A square and a rectangle have the same area. The rectangle measures 18 cm × 32 cm. Find the perimeter of the square.
= 18 × 32
= 576 cm²
Side of square
= √576
= 24 cm
Perimeter
= 4 × 24
= 96 cm
Question 28: The diagonals of a square intersect at O. If AO = (2x + 3) cm and the side of the square is 10√2 cm, find the value of x.
= 10√2 × √2
= 20 cm
Since diagonals bisect each other,
AO = 10 cm
Therefore,
2x + 3 = 10
2x = 7
x = 3.5
Question 29: The ratio of the diagonals of two squares is 5 : 8. If the area of the smaller square is 450 cm², find the area of the larger square.
Therefore,
Area ratio
= 25 : 64
Let the larger area be A.
450 : A = 25 : 64
A = 450 × 64 ÷ 25
= 1152 cm²
Question 30: The diagonals of a rhombus are (3x + 5) cm and (2x + 9) cm. If one diagonal is 8 cm longer than the other, find the value of x and the lengths of the diagonals.
(3x + 5) − (2x + 9) = 8
x − 4 = 8
x = 12
Therefore,
First diagonal = 3(12) + 5 = 41 cm
Second diagonal = 2(12) + 9 = 33 cm
Question 31: A rhombus has side 25 cm and one diagonal 14 cm. Find the length of the other diagonal.
Let half of the other diagonal be x cm.
Using Pythagoras’ theorem,
25² = 7² + x²
625 = 49 + x²
x² = 576
x = 24
Other diagonal = 48 cm
Question 32: The perimeter of a rhombus is 104 cm. One diagonal is 20 cm. Find the length of the other diagonal.
= 104 ÷ 4
= 26 cm
Half of one diagonal
= 10 cm
Let half of the other diagonal be x cm.
Using Pythagoras’ theorem,
26² = 10² + x²
676 = 100 + x²
x² = 576
x = 24
Other diagonal = 48 cm
Question 33: The diagonals of a rhombus are in the ratio 5 : 12. If each side is 26 cm, find the lengths of the diagonals.
Half of the diagonals are:
5x⁄2 and 6x.
Using Pythagoras’ theorem,
26² = (5x⁄2)² + (6x)²
676 = 25x²⁄4 + 36x²
676 = 169x²⁄4
x² = 16
x = 4
Diagonals:
20 cm and 48 cm
Question 34: The area of a rhombus is 840 cm² and its perimeter is 116 cm. Find the lengths of its diagonals.
= 116 ÷ 4
= 29 cm
Let the diagonals be d₁ and d₂.
Area:
½d₁d₂ = 840
d₁d₂ = 1680
Also,
(d₁⁄2)² + (d₂⁄2)² = 29²
d₁² + d₂² = 3364
Now,
(d₁ + d₂)²
= d₁² + d₂² + 2d₁d₂
= 3364 + 3360
= 6724
d₁ + d₂ = 82
The numbers whose sum is 82 and product is 1680 are:
42 and 40.
Question 35: In an isosceles trapezium, the parallel sides measure 18 cm and 42 cm. If each non-parallel side measures 15 cm, find the height of the trapezium.
= 42 − 18
= 24 cm
Half of the difference
= 12 cm
Using Pythagoras’ theorem,
Height² = 15² − 12²
= 225 − 144
= 81
Height = 9 cm
Question 36: The area of a trapezium is 324 cm². The parallel sides are in the ratio 5 : 13 and the height is 18 cm. Find the lengths of the parallel sides.
Area
= ½ × (5x + 13x) × 18
324 = 9 × 18x
324 = 162x
x = 2
Parallel sides
= 10 cm and 26 cm
Question 37: The area of a trapezium is 420 cm². Its height is 15 cm and one parallel side is 8 cm longer than the other. Find the lengths of the parallel sides.
Longer side = (x + 8) cm.
420 = ½(x + x + 8) × 15
420 = 15(x + 4)
x + 4 = 28
x = 24
Longer side = 32
Question 38: The diagonals of a kite are in the ratio 3 : 5. If its area is 540 cm², find the lengths of the diagonals.
540 = ½ × 3x × 5x
1080 = 15x²
x² = 72
x = 6√2
Therefore,
Diagonals
= 18√2 cm and 30√2 cm
Question 39: An isosceles trapezium has equal diagonals of length 26 cm. The height is 24 cm. If the difference between the parallel sides is 20 cm, find the lengths of the parallel sides.
Difference
= 20 cm
Half the difference
= 10 cm
Using Pythagoras’ theorem,
Horizontal distance from one end of the shorter base to the opposite end of the longer base
= √(26² − 24²)
= √100
= 10 cm
Hence,
(a + b)/2 = 10
a + b = 20
But
b − a = 20
Solving,
a = 10
b = 30
Question 40: A quadrilateral has all four sides equal. One diagonal is 48 cm long and the other is 20 cm long. Identify the quadrilateral and find its perimeter.
The diagonals bisect each other at right angles.
Half of the diagonals are:
24 cm and 10 cm.
Using Pythagoras’ theorem,
Side² = 24² + 10²
= 576 + 100
= 676
Side = 26 cm
Perimeter = 4 × 26 = 104 cm
Question 41: A parallelogram and a rhombus each have a perimeter of 80 cm. One side of the parallelogram is 12 cm. Find the length of the adjacent side of the parallelogram and the side of the rhombus.
2(a + b) = 80
a + b = 40
12 + b = 40
b = 28 cm
For the rhombus,
Each side = 80 ÷ 4
= 20 cm
Adjacent side of the parallelogram = 28 cm, Side of the rhombus = 20 cm.
Question 42: The diagonals of a rhombus are 30 cm and 16 cm. A square has the same perimeter as the rhombus. Find the area of the square.
15 cm and 8 cm.
Using Pythagoras’ theorem,
Side² = 15² + 8²
= 225 + 64
= 289
Side = 17 cm
Perimeter of the rhombus
= 68 cm
Side of the square
= 68 ÷ 4
= 17 cm
Area = 17²
= 289 cm²
Question 43: A rectangle has dimensions 24 cm × 18 cm. A rhombus has the same perimeter as the rectangle. If one diagonal of the rhombus is 24 cm, find the length of the other diagonal.
= 2(24 + 18)
= 84 cm
Therefore, each side of the rhombus
= 84 ÷ 4
= 21 cm
Half of the known diagonal
= 12 cm
Let half of the other diagonal be x cm.
Using Pythagoras’ theorem,
21² = 12² + x²
441 = 144 + x²
x² = 297
x = 3√33
Other diagonal
= 6√33 cm
Question 44: A rectangle has length 30 cm and breadth 16 cm. A square has the same diagonal as the rectangle. Find the area of the square.
= √(30² + 16²)
= √1156
= 34 cm
Let the side of the square be s cm.
s√2 = 34
s = 17√2 cm
Area = (17√2)²
= 578 cm²
Question 45: A rectangle and a parallelogram have the same perimeter of 72 cm. The rectangle measures 20 cm × 16 cm. One side of the parallelogram is 14 cm. Find the length of the adjacent side of the parallelogram and compare the areas of the two figures if the height of the parallelogram is 12 cm.
= 72 cm
Therefore,
2(a + b) = 72
a + b = 36
Given,
a = 14 cm
b = 36 − 14 = 22 cm
Area of the rectangle
= 20 × 16
= 320 cm²
Area of the parallelogram
= Base × Height
= 22 × 12
= 264 cm²
Difference in areas
= 320 − 264
= 56 cm²
What are Quadrilaterals?
A quadrilateral is a closed two-dimensional polygon formed by four sides, four vertices and four angles. The sum of the interior angles of every quadrilateral is 360°. Quadrilaterals are classified into different types, such as square, rectangle, parallelogram, rhombus, trapezium and kite, based on the lengths of their sides, the measures of their angles and the properties of their diagonals.
Parts of a Quadrilateral
A quadrilateral consists of the following parts:
- Sides- The four line segments that form the boundary of the quadrilateral.
- Vertices- The four points where two sides meet.
- Angles- The four interior angles formed by the adjacent sides.
- Diagonals- The two line segments joining opposite vertices.
- Properties-
A quadrilateral has 4 sides, 4 vertices, 4 angles and 2 diagonals.
Every quadrilateral is a polygon, but not every polygon is a quadrilateral.
Sum of Interior Angles of a Quadrilateral
Sum of Interior Angles = 360°
This property is true for all quadrilaterals, whether they are regular or irregular.
Types of Quadrilaterals
Quadrilaterals are classified according to the lengths of their sides, the measures of their angles and the properties of their diagonals. The common types are:
- Square
- Rectangle
- Parallelogram
- Rhombus
- Trapezium
- Kite
Each type has its own unique set of properties.
Properties of Parallelogram
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. The properties of parallelogram are:
- Opposite sides are equal
- Opposite sides are parallel
- Opposite angles are equal
- Adjacent angles are supplementary
- Diagonals bisect each other
Properties of Rectangle
A rectangle is a parallelogram in which each interior angle is 90°. The properties of Rectangle are:
- Opposite sides are equal and parallel
- All four angles are 90°
- Diagonals are equal
- Diagonals bisect each other
Properties of Squaure
A square is a rectangle with all four sides equal. The properties of Square are:
- All four sides are equal
- Opposite sides are parallel
- Each interior angle is 90°
- Diagonals are equal
- Diagonals bisect each other
- Diagonals are perpendicular
- Diagonals bisect the opposite angles
Properties of Rhombus
A rhombus is a parallelogram with all four sides equal. The properties of Rhombus are:
- All four sides are equal
- Opposite sides are parallel
- Opposite angles are equal
- Adjacent angles are supplementary
- Diagonals bisect each other
- Diagonals are perpendicular
- Diagonals bisect the opposite angles
Properties of Trapezium
- Exactly one pair of opposite sides is parallel
- The parallel sides are called the bases
- The non-parallel sides are called the legs
- Non-parallel sides are equal
- Base angles are equal
- Diagonals are equal
Special case
Properties of Kite
- Two pairs of adjacent sides are equal
- One pair of opposite angles is equal.
- Diagonals are perpendicular
- One diagonal bisects the other.
- One diagonal bisects a pair of opposite angles
Relationship Between Different Quadrilaterals
- Every square is a rectangle
- Every square is a rhombus
- Every rectangle is a parallelogram.
- Every rhombus is a parallelogram.
- Every parallelogram is a quadrilateral
- A rectangle is not necessarily a square
- A rhombus is not necessarily a square
- A parallelogram is not necessarily a rectangle or a rhombus.
However
Summary of Quadrilaterals
| Quadrilateral | Key Properties |
|---|---|
| General Quadrilateral | Has 4 sides, 4 vertices, 4 angles and 2 diagonals. The sum of the interior angles is 360°. |
| Parallelogram | Opposite sides are equal and parallel, opposite angles are equal, adjacent angles are supplementary, and diagonals bisect each other. |
| Rectangle | Opposite sides are equal and parallel, all angles are 90°, diagonals are equal and bisect each other. |
| Square | All sides are equal, all angles are 90°, diagonals are equal, perpendicular, bisect each other and bisect the opposite angles. |
| Rhombus | All sides are equal, opposite sides are parallel, opposite angles are equal, diagonals are perpendicular and bisect each other as well as the opposite angles. |
| Trapezium | Has one pair of opposite sides parallel. In an isosceles trapezium, the non-parallel sides and diagonals are equal. |
| Kite | Has two distinct pairs of adjacent equal sides. One pair of opposite angles is equal, diagonals are perpendicular, and one diagonal bisects the other. |
Common Mistakes
- Confusing the Properties of Different Quadrilaterals:
Students often assume that every parallelogram has equal diagonals or that every quadrilateral with equal sides is a square. Learn the defining properties of each quadrilateral before applying any theorem. - Using the Wrong Angle Sum:
Students sometimes use the interior angle sum formula for polygons instead of remembering that the sum of the interior angles of every quadrilateral is always 360°.
Formula:
Sum of Interior Angles = 360° - Confusing the Properties of Diagonals:
Students often assume that the diagonals of all quadrilaterals behave the same way. For example, the diagonals of a rectangle are equal, those of a rhombus are perpendicular bisectors, those of a square are both equal and perpendicular bisectors, while those of a general parallelogram only bisect each other. - Using the Wrong Area Formula:
Students sometimes interchange the area formulas of different quadrilaterals. Always identify the figure first before selecting the appropriate formula.
Formulae:
Rectangle = Length × Breadth
Parallelogram = Base × Height
Rhombus = ½ × (Diagonal₁ × Diagonal₂)
Trapezium = ½ × (Sum of Parallel Sides) × Height
Kite = ½ × (Diagonal₁ × Diagonal₂) - Ignoring Perpendicular Diagonals in a Rhombus or Kite:
Students often forget that the diagonals of a rhombus and a kite are perpendicular. This creates right triangles, making Pythagoras’ theorem useful for finding unknown sides or diagonals.
Practice Questions
Question 1: A quadrilateral has all four sides equal. One diagonal is 24 cm and the other is 70 cm. Identify the quadrilateral and find its perimeter.
Question 2: An isosceles trapezium has parallel sides measuring 22 cm and 40 cm. If each non-parallel side measures 15 cm, find its height and area.
Question 3: A rectangle has dimensions 21 cm × 20 cm. A parallelogram has the same area and the same base as the rectangle. If one side of the parallelogram is 25 cm, find its height and perimeter.
Question 4: A parallelogram has a perimeter of 76 cm. One side is 15 cm longer than the adjacent side. Find the lengths of its sides.
Question 5: In a quadrilateral ABCD, the measures of the interior angles are ∠A = 5x, ∠B = 2x + 15°, ∠C = 3x, and ∠D = 2x + 5°. Find the value of x and the measure of each angle.
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