Class 8 Maths Chapter 11 Parallelograms Questions and Answers Kerala Syllabus

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SCERT Class 8 Maths Chapter 11 Solutions Parallelograms

Class 8 Kerala Syllabus Maths Solutions Chapter 11 Parallelograms Questions and Answers

Parallelograms Class 8 Questions and Answers Kerala Syllabus

Sides and Angles (Page 178)

Question 1.
The figures below are combinations of three, four, and five equal rhombuses. Draw each and colour it.

Answer:
1. Draw a circle of radius 2 cm (if any).
Divide the centre of the circle into angles of 60° each.
These lines meet the circle at the points A, B, C, D, E, and F.

Draw arcs of 2 cm (equal to radius) from A and B to get G.
Similarly, find H and I.
Draw the required parts and rub off unwanted parts.

2. Draw a circle of radius 2 cm (if any).
Mark the points A, B, C, D, E, F, G, and H on the circle by making 45° angles at the centre.
Draw an arc of 2 cm (if any), A and B to get I.
Similarly, find J, K, and L.
We get the required figure.

3. Draw AC, 4 cm long, and mark its midpoint B.
Since all are rhombuses, ABI is a equilateral triangle.
Its angles are 60° each.

In the rhombus BCDJ,
JBC = JDC = 60°
BCD = BJD = 120°
In the rhombus BJFI,
IBJ = IFJ = 60°
BJF = FIB =120°.
Draw each rhombus and complete the pattern.

Question 2.
This picture is made up of three equal parallelograms and an equilateral triangle. Draw and colour it.

Answer:
The figure is an optical illusion of a 3D cube or hexagon composed of three equal parallelograms and one equilateral triangle.

• Draw an equilateral triangle (all sides equal, all angles 60°).
• On each of the three sides of the triangle, draw a parallelogram.
• To make the figure symmetrical, the angles of the parallelograms meeting at the triangle’s vertices should be calculated such that they fit perfectly.
• Usually, the parallelograms are identical to the triangle in terms of side length.

Question 3.
Draw and colour this picture made up of four equal parallelograms and four equal right triangles:

Answer:
Construction:

• Draw a central square or rectangle (formed by the space or the meeting points).
• Attach right-angled triangles to the sides.
• Extend the pattern by attaching the parallelograms to the hypotenuse or legs of the triangles, depending on the specific pattern shown in your textbook.
• Without seeing the exact visual pattern in the book, the key is to ensure the right angles are exactly 90° and opposite sides of the parallelograms are equal.

Diagonals (Pages 182-183)

Question 1.
Draw a parallelogram with lengths of diagonals 8 centimetres and 6 centimetres and the angle between them 60°.
Answer:
Construction Steps:
Draw a line segment AC = 8 cm.
Mark the midpoint O of AC (so AO = 4 cm).
Through O, draw a line XY making an angle of 60° with AC.
On this line XY, mark points B and D on opposite sides of 0 such that OB = 3 cm and OD = 3 cm.
(since the second diagonal is 6 cm and diagonals bisect each other).
Join the points A, B, C, and D to form the parallelogram ABCD.

Question 2.
Prove that in a parallelogram, the line joining the midpoints of two opposite sides is parallel to the other two sides:

Answer:
(i) Let ABCD be a parallelogram.

Let P be the midpoint of side AB, and Q be the midpoint of side CD.
Since ABCD is a parallelogram, AB || CD and AB = CD.
Since P and Q are midpoints,
AP = \(\frac {1}{2}\)AB and DQ = \(\frac {1}{2}\)CD.
Therefore, AP = DQ and AP || DQ.
A quadrilateral with one pair of opposite sides equal and parallel is a parallelogram.
So, APQD is a parallelogram.
This implies PQ || AD.
Thus, the line joining the midpoints is parallel to the other two sides.

(ii) Let ABCD be a parallelogram.

Let P be the midpoint of side AD and Q be the midpoint of side BC.
Since ABCD is a parallelogram, AD || BC and AD = BC.
Since P and Q are midpoints,
AP = \(\frac {1}{2}\)AD and BQ = \(\frac {1}{2}\)BC.
Therefore, AP = BQ and AP || BQ.
A quadrilateral with one pair of opposite sides equal and parallel is a parallelogram.
So, ABQP is a parallelogram.
This implies PQ || AB.
Thus, the line joining the midpoints is parallel to the other two sides.

Question 3.
The picture shows a parallelogram and the lines joining the midpoint of opposite sides:

(i) Prove that these lines bisect each other.
(ii) Prove that an angle between these lines is equal to an angle between the sides of the parallelogram.
Answer:
(i) Let the parallelogram be ABCD.
EF: joining midpoints of AB and CD
GH: joining midpoints of BC and AD

In parallelogram ABCD:
Opposite sides are parallel and equal:
AB || CD, BC || AD
E and F are midpoints,
AE = EB, CF = FD
Because AB || CD, the segment EF is a mid-segment between two parallel lines.
Similarly, GH is a mid-segment of the pair AD || BC.
The segment joining midpoints of one pair of opposite sides is parallel to the other pair of opposite sides.
Thus EF || AD || BC; GH || AB || CD
Because EF is parallel to BC and AD, and GH is parallel to AB and CD, the shape EFGH is a parallelogram (by opposite sides being parallel).
In every parallelogram, the diagonals bisect each other.
Therefore, X is the midpoint of both EF and GH.

(ii) We already saw:
EF || AD || BC
GH || AB || CD
So the angle between the lines EF and GH is equal to the angle between the pairs of parallel sides:
∠FXG = ∠HAE
Because corresponding angles between parallel lines are equal.
Thus, the angle between the mid-point joining lines equals an interior angle of the parallelogram.
Hence proved.

Question 4.
Draw a parallelogram with lengths of diagonals 6.5 centimetres and 4.5 centimetres and the angle between them 70°.
Answer:

Draw a line segment AC = 6.5 cm.
Mark its midpoint O (at 3.25 cm).
Through O, draw a line making an angle of 70° with AC.
On this new line, mark points B and D such that OB = 2.25 cm and OD = 2.25 cm (half of 4.5 cm).
Join A, B, C, and D.

Question 5.
Prove that if the diagonals of a parallelogram are of the same length, then it is a rectangle.
Answer:

Let ABCD be a parallelogram where diagonals AC = BD.
Consider the triangles ∆ABC and ∆BAD.
AB = AB (Common side).
BC = AD (Opposite sides of a parallelogram are equal).
AC = BD (Given).
By the SSS congruence rule,
∆ABC ≅ ∆BAD.
Therefore, their corresponding angles are equal:
∠ABC = ∠BAD
Since ABCD is a parallelogram, adjacent angles sum to 180° (∠ABC + ∠BAD = 180°).
Since they are equal,
2∠ABC = 180°
∴ ∠ABC = 90°.
A parallelogram with a right angle is a rectangle.

Question 6.
Draw a rectangle with a length of the diagonals of 6 centimetres and the angle between them 60°.
Answer:
Construction Steps:
Draw a line segment AC = 6 cm, and mark its midpoint O.
Through O, draw a line making an angle of 60° with AC.
On this line, mark points B and D such that OB = 3 cm and OD = 3 cm
Since the diagonals of a rectangle are equal and bisect each other, half of 6 cm is 3 cm.
Join A, B, C, and D.
You will get the required rectangle.

Area (Pages 187-188)

Question 1.
Draw a parallelogram with lengths of sides 5 centimetres and 6 centimetres, and an area of 25 square centimetres.
Answer:
Area of a parallelogram = one side × distance to the opposite side.
Since 5 × distance to the opposite side is 25 sq.cm.
Distance to the opposite side is 5 cm.
Draw a square ABCD of side 5 cm.
Draw an arc of radius 6 cm with centre A to cut DC at E.
Length of radius 6cm with centre B and arc of radius 5 cm with centre E meet at F.
Draw BC and FC.
ABFE is the required parallelogram.

Question 2.
Draw a parallelogram with area 25 square centimetres and perimeter 24 centimetres.
Answer:
Area of the parallelogram is to be 25 cm².
The product of one side and the distance to the opposite side is 25 cm².
One side and the distance to the opposite side can be 5 cm each.
The other side is 12 – 5 = 7 cm.
Draw a square with a side of 5cm and draw an arc of radius 7 cm with A as centre, such that it intersects the side CD.
Mark the point E and join AE.
Draw an arc of radius 7 cm with B as centre, and draw another arc of radius 5 cm with E as centre. Two arcs meet at the point P.
Join BF and CF.

Question 3.
The picture shows the parallelogram formed by the intersection of two pairs of parallel lines:

What is the area of the parallelogram? And the perimeter?
Answer:
One side of the parallelogram = 4 cm
Distance to the opposite side = 3 cm
Area of the parallelogram = 4 × 3 = 12 sq.cm

BC × DE = 12
BC × 2 = 12
BC = 6, AD = 6
Perimeter = 4 + 6 + 4 + 6 = 20 cm.

Question 4.
Two sides of a parallelogram are 12 centimetres and 10 centimetres, and the distance between the shorter sides is 6 centimetres.
(i) What is the area of the parallelogram?
(ii) What is the distance between the longer sides?
Answer:
Let the parallelogram have:
Longer side = 12 cm
Shorter side = 10 cm
Distance (height) between shorter sides = 6 cm
(i) Area of the parallelogram
Area = base × height
(If we take the shorter side (10 cm) as the base, its height is 6 cm)
Area = 10 × 6 = 60 cm²

(ii) Let the distance between the 12 cm sides be (h).
Since area = base × height for any pair of opposite sides:
60 = 12 × h
⇒ h = \(\frac {60}{12}\)
⇒ h = 5 cm

Rhombuses (Page 190)

Question 1.
Draw a square of area 4\(\frac {1}{2}\) square centimetres.
Answer:
\(\frac {1}{2}\)d² = 4\(\frac {1}{2}\)
⇒ d² = 2 × 4\(\frac {1}{2}\) = 9
⇒ d = 3 cm

Draw a circle of radius 3 cm.
Draw the diameter AC and construct the perpendicular bisector of AC, which meets the circle at B and D.
Join AC, AD, CD, and BD to get the square ABCD.

Question 2.
Draw a rhombus of area 9 square centimetres, which is not a square.
Answer:

Area = \(\frac {1}{2}\) × d₁ × d₂ = 9
d₁ × d₂ = 18
So, 1 × 18 = 18
2 × 9 = 18
3 × 6 = 18
Let d₁ = 1 and d₂ = 18
The product of the diagonals of the rhombus should be 18.
There are three pairs of natural numbers: 18, 1; 9, 2, and 6, 3 as the lengths of the diagonals.
You can find more pairs of fractions.
A rhombus with diagonals 6 cm and 3 cm is drawn here.
You draw the other two.

Draw AC, 6 cm long.
Draw its perpendicular bisector and mark points B and D on it such that the distance from AC is 1.5 cm each.
Join AB, BC, CD, and AD to complete the rhombus.

Question 3.
The picture shows a quadrilateral drawn by joining the midpoints of the diagonals of a rhombus:

(i) Is this quadrilateral also a rhombus? Why?
(ii) The area of the small quadrilateral is 3 square centimetres. What is the area of the large rhombus?
Answer:
(i) The diagonals of the rhombus intersect at O, and they bisect each other at right angles.

OA = OC
\(\frac{\mathrm{OA}}{2}=\frac{\mathrm{OC}}{2}\)
OQ = OS
Similarly, since OB = OD
i.e., OR = OP
The diagonals of PQRS bisect each other.
(Since OQ = OS and OR = OP)
Since the diagonal AC is perpendicular to BD, the diagonals of PQRS are also mutually perpendicular bisectors.
Therefore, PQRS is a rhombus.

(ii) Area of quadrilateral PQRS = 3 sq.cm
\(\frac {1}{2}\)d₁d₂ = 3 sq. cm
Area of rhombus ABCD = \(\frac {1}{2}\) × 2d₁ × 2d₂
= \(\frac {1}{2}\) × 4d₁d₂
= 4 × \(\frac {1}{2}\)d₁d₂
= 4 × 3
= 12 sq. cm

Question 4.
The sides of a rhombus are 10 centimetres long, and one of its diagonals is 16 centimetres long
(i) What is the length of the other diagonal?
(ii) What is the area of the rhombus?
(iii) What is the distance between the opposite sides?
Answer:
(i) d₁ = 16 cm

We have a rhombus with a side of 10 cm and a diagonal BD = 16 cm
We know that the diagonals of a rhombus bisect each other at 90°.
BO = OD = 8 cm
In ∆AOB, by Pythagoras theorem,
AO² + BO² = AB²
⇒ AO² + 8² = 10²
⇒ AO² = 100 – 64 = 36
⇒ AO = 6 cm [By above property]
Hence, AC = 6 + 6 = 12 cm

(ii) Area = \(\frac {1}{2}\)d1d2
= \(\frac {1}{2}\) × 16 × 12
= 96 cm²

(iii) The distance between opposite sides is the height.
Area = Base × Height (∵ Base is the side length)
96 = 10 × Height
Height = \(\frac {96}{10}\)
Height = 9.6 cm

Class 8 Maths Chapter 11 Kerala Syllabus Parallelograms Questions and Answers

Class 8 Maths Parallelograms Questions and Answers

Question 1.
Which of the following is a defining property of every parallelogram?
(A) Diagonals are equal
(B) All sides are equal
(C) Opposite sides are parallel
(D) All angles are 90°
Answer:
(C) Opposite sides are parallel
This is the basic definition of a parallelogram.

Question 2.
If the adjacent sides of a parallelogram are 8 cm and 6 cm, what is its perimeter?
(A) 14 cm
(B) 28 cm
(C) 48 cm
(D) 24 cm
Answer:
(B) 28 cm
Perimeter = 2(8 + 6)
= 2 × 14
= 28 cm

Question 3.
A parallelogram has a base of 10 cm, and the distance to the opposite side (height) is 6 cm. What is its area?
(A) 60 sq.cm
(B) 30 sq.cm
(C) 16 sq.cm
(D) 32 sq.cm
Answer:
(A) 60 sq. cm
Area = Base × Height
= 10 × 6
= 60

Question 4.
In a parallelogram, the diagonals:
(A) Are always equal
(B) Bisect each other
(C) Are always perpendicular
(D) Bisect the angles
Answer:
(B) Bisect each other
A key property of parallelogram diagonals.

Question 5.
If the diagonals of a parallelogram are equal in length, the figure is a:
(A) Rhombus
(B) Trapezium
(C) Rectangle
(D) Kite
Answer:
(C) Rectangle
If diagonals bisect each other and are equal, it’s a rectangle.

Question 6.
Calculate the area of a rhombus whose diagonals are 10 cm and 12 cm.
(A) 120 sq.cm
(B) 60 sq. cm
(C) 100 sq. cm
(D) 22 sq. cm
Answer:
(B) 60 sq. cm
Area of Rhombus = \(\frac {1}{2}\) × d₁ × d₂
= \(\frac {1}{2}\) × 10 × 12
= 60

Question 7.
A square has a diagonal of length 8 cm. What is its area?
(A) 64 sq.cm
(B) 16 sq.cm
(C) 32 sq.cm
(D) 48 sq.cm
Answer:
(C) 32 sq.cm
Area of Square = \(\frac {1}{2}\) × diagonal²
= \(\frac {1}{2}\) × 8²
= \(\frac {1}{2}\) × 64
= 32

Question 8.
If one angle of a parallelogram is 70°, what is the measure of the angle opposite to it?
(A) 110°
(B) 70°
(C) 90°
(D) 20°
Answer:
(B) 70°
Opposite angles in a parallelogram are equal.

Question 9.
If one angle of a parallelogram is 60°, what is the measure of an adjacent angle?
(A) 60°
(B) 30°
(C) 120°
(D) 90°
Answer:
(C) 120°
Adjacent angles sum to 180°.
180° – 60° = 120°

Question 10.
A diagonal divides a parallelogram into two triangles. If the area of the parallelogram is 50 sq. cm, what is the area of one of these triangles?
(A) 50 sq.cm
(B) 100 sq.cm
(C) 25 sq.cm
(D) 10 sq.cm
Answer:
(C) 25 sq.cm
A diagonal splits a parallelogram into two equal triangles.
\(\frac {50}{2}\) = 25

Question 11.
Which quadrilateral has diagonals that are perpendicular bisectors of each other?
(A) Rectangle
(B) Parallelogram
(C) Rhombus
(D) Trapezium
Answer:
(C) Rhombus
Diagonals of a rhombus are perpendicular bisectors.

Question 12.
The area of a parallelogram is given by:
(A) Base × Height
(B) \(\frac {1}{2}\) × Base × Height
(C) Product of adjacent sides
(D) Product of diagonals
Answer:
(A) Base × Height

Question 13.
Read the following statements:
Statement I: All squares are rhombuses.
Statement II: All rhombuses are squares.
Choose the correct option:
(A) Statement I is true, Statement II is false.
(B) Statement I is false, Statement II is true.
(C) Both statements are true.
(D) Both statements are false.
Answer:
(A) Statement I is true, Statement II is false
A square has all the properties of a rhombus (equal sides), but a rhombus doesn’t necessarily have 90° angles like a square.

Question 14.
Read the following statements:
Statement I: The area of a square is half the square of its diagonal \(\frac {1}{2}\)d².
Statement II: A square is a rhombus with equal diagonals.
Choose the correct option:
(A) Only I is true.
(B) Only II is true.
(C) Both are true, and II explains I.
(D) Both are true, but II does not explain I.
Answer:
(C) Both are true, and II explains I
Since a square is a rhombus with equal diagonals, we can use the rhombus formula \(\frac {1}{2}\) × d₁ × d₂, which becomes \(\frac {1}{2}\)d².

Question 15.
A quadrilateral where only one pair of opposite sides is parallel is called a:
(A) Parallelogram
(B) Rhombus
(C) Trapezium
(D) Rectangle
Answer:
(C) Trapezium

Question 16.
To construct a unique parallelogram, which of the following sets of measurements is sufficient?
(A) Lengths of two adjacent sides
(B) Lengths of two adjacent sides and the included angle
(C) Length of one side
(D) Measure of one angle
Answer:
(B) Lengths of two adjacent sides and the included angle
Just sides aren’t enough; the angle determines the shape/area.

Question 17.
The perimeter of a rhombus is 40 cm. What is the length of one side?
(A) 20 cm
(B) 5 cm
(C) 10 cm
(D) 8 cm
Answer:
(C) 10 cm
A rhombus has 4 equal sides.
\(\frac {40}{4}\) = 10

Question 18.
In the figure of a parallelogram, if the distance between the longer sides (10 cm) is 4 cm, what is the area?
(A) 14 sq. cm
(B) 20 sq. cm
(C) 40 sq. cm
(D) 80 sq. cm
Answer:
(C) 40 sq. cm
Area = Side × Distance
= 10 × 4
= 40

Question 19.
Read the following statements about a Rhombus:
Statement I: The diagonals are of equal length.
Statement II: The area is half the product of the diagonals.
Choose the correct option:
(A) Statement I is true, Statement II is false.
(B) Statement I is false, Statement II is true.
(C) Both are true.
(D) Both are false.
Answer:
(B) Statement I is false, Statement II is true
Diagonals of a rhombus are usually unequal (unless it’s a square).
The area formula is correct.

Question 20.
A parallelogram has sides 12 cm and 8 cm. The distance between the 12 cm sides is 4 cm. What is the distance between the 8 cm sides?
(A) 6 cm
(B) 4 cm
(C) 12 cm
(D) 8 cm
Answer:
(A) 6 cm
Area is constant.
12 × 4 = 48.
So, 8 × h = 48
∴ h = 6

Question 21.
Draw a rhombus of diagonals 5.5 cm and 3 cm in your notebook.
Answer:
Draw a line of length 5.5 cm, and find its midpoint by drawing the perpendicular bisector.
Mark the points on the upper and lower parts of the bisector line at a distance of 1.5 cm from the intersecting point of the first line and the perpendicular bisector.
Join these points to the end of the first line.

Question 22.
Draw these figures
1. Two equal rhombuses

2. Parallelograms on two sides of a square

Answer:
1. From the figure two sides of ∆BCD are equal, angles opposite these sides are also equal.
We can calculate them as 50° each.
In the same way, find other angles in the figure.

Draw a line BD vertically, 3 cm long.
At D, draw angles of 50° on both sides.
At B, also draw angles of 50° on both sides.
Then we get a rhombus ABCD.
Extend BC to G such that BC = CG, and extend DC to E such that DC = CE.
Draw GE. Draw an angle of 50° at G and E to find F.

2. Draw a square of side 3 cm.
And draw two parallelograms with sides 3 cm, 2 cm, and an angle between them of 45°, on the sides of the square.

Question 23.
In rhombus PQRS, PR = 7 cm and QS = 5 cm. Construct a rhombus PQRS.
Answer:

Question 24.
In the figure, ABCD is a parallelogram. D = 80°. Find all other angles?

Answer:
ABCD is a parallelogram
Opposite angles are equal.
B = 80°.
The sum of the angles on the same side is 180°.
∠A + ∠B = 180°.
∠A = 180° – 80° = 100°
And ∠C = 100° (opposite angles are equal).

Question 25.
In the figure, ABCD is a parallelogram. Find x, y, z.

Answer:
∠Y = 112° (opposite angles are equal)
In ADC, ∠x + ∠y + 40 = 180° (sum of angles in a triangle)
∠x + 112° + 40° = 180°
∠x = 180° – 152° = 28°
∠z = 28° (transversal alternate interior angles are equal).

Question 26.
In parallelogram ABCD, the diagonals AC and BD intersect at O. AC = 6.5 cm, BD = 7 cm, and ∠AOB = 100°. Construct the parallelogram.
Answer:

Question 27.
The diagonals of a rhombus are of lengths 16 cm and 12 cm. What is its perimeter?
Answer:

In right-angled AOB,
AO = 8 cm, BO = 6 cm, AOB = 90°
AB² = AO² + BO²
= 8² + 6²
= 64 + 36
= 100
Side, AB = 10 cm
Perimeter = 4 × 10 = 40 cm.

Question 28.
Draw the following patterns.
(a) 6 equal rhombuses

(b) 3 equal rhombuses

(c) Two equal rhombuses on the sides of a square

(d) 4 equal rhombuses

(e) Parallelograms on two sides of a rectangle of sides 6 cm and 3 cm.

Answer:
(a) Draw a circle of radius 2 cm with centre O.
Divide the centre of the circle into angles of 60° each.
These lines meet the circle at the points A, B, C, D, E, and F.
Draw an arc of 2 cm from A and B to get G.
In the same way, find H, I, J, K, L.
Draw needed part.

(b) Draw a square of side 4 cm.
Draw rhombuses with a side of 4 cm and an angle of 30° on the top and bottom sides of the square.
Complete the figure.

(c) Angles around the point at which three rhombuses joined together are 120° each.
Since one angle of the rhombus is 120°, another angle is 60°.
Draw three rhombuses with a side of 4 cm and an angle of 60°.
Complete the figure.

(d) Draw a semicircle of radius 2 cm with centre O.
Divide the centre of the circle into angles of 45° each.
These lines meet the circle at the points A, B, C, D, and E.
Draw an arc of 2 cm from A and B to get F.
In the same way, find G, H, and I.
Complete the figure.

(e) Draw a rectangle of length 6 cm and breadth 3 cm.
Draw two rhombuses on both sides of the rectangle, which makes angle 45° and 135° with length and breadth, respectively.

Question 29.
In a parallelogram ABCD, find x, y, z from the adjoining figure.

Answer:
ABCD is a parallelogram,
∠C = 45° (Opposite angles are equal)
∠C + Z = 180° (linear pair)
Z = 180° – 45° = 135°
45° + Y = 180° (Sum of the angles on the same side is 180°)
Y = 180° – 45° = 135°
Since Y = 135°
X = 135° (Opposite angles are equal)

Question 30.
Which of the following has a greater area: a parallelogram with one side 12 cm and a distance between parallel sides of 6 cm, or a square with diagonals of 12 cm each?
Answer:
Area of parallelogram = \(\frac {1}{2}\) × b × h
= \(\frac {1}{2}\) × 12 × 12
= 72 cm²
Area of square = \(\frac {1}{2}\) × d²
= \(\frac {1}{2}\) × 12²
= \(\frac {1}{2}\) × 144
= 72 cm²
Area of both are equal.

Question 31.
Find the area of a rhombus with one side of 6 cm and one diagonal of 6 cm.
Answer:
The diagonals of a rhombus bisect each other perpendicularly.

In ∆AOB, AO² = AB² – BO²
= 6² – 3²
= 36 – 9
= 25
AO = √25 = 5 cm
AC = 10 cm
Area of the rhombus = \(\frac {1}{2}\) × d₁ × d₂
= \(\frac {1}{2}\) × 6 × 10
= 30 sq. cm

Question 32.
In the pictures given below, which one has more area?

Answer:
A rectangle has the maximum area among the parallelograms with the same sides.

Question 33.
The ratio of the two adjacent sides of a parallelogram is 3 : 2. The distance between the longer sides is 10 cm. If the area is 900 cm², find the sides of the parallelogram.
Answer:
Let the sides be 3x and 2x

3x × 10 = 900
⇒ 30x = 900
⇒ x = 30 cm
Sides are: 3 × 30 = 90 cm; 2 × 30 = 60 cm

Question 34.
If one side of a parallelogram is ‘a’ and the height of that side is h, prove that the area = ah.
Answer:
In the figure, ABCD is a parallelogram.
AB = a

The perpendicular distance from D to AB = h
By drawing the diagonal BD, we can divide the parallelogram into two equal triangles.
∆ABD and ∆BCD are equal triangles; their areas are equal.
i.e, the area of the parallelogram is two times of the area of ∆ABD.
Area of ∆ABD = \(\frac {1}{2}\) × a × h
Area of the parallelogram ABCD = 2 × \(\frac {1}{2}\) × ah = ah

Question 35.
PQRS is a rhombus. If the diagonals are 8 cm and 9 cm each, compute the area.

Answer:
Let the diagonals of the rhombus be d₁ and d₂
Area = \(\frac {1}{2}\) × d₁ × d₂
= \(\frac {1}{2}\) × 8 × 9
= 36 cm²

Question 36.
The perimeter of a rhombus is 40 cm. If the length of one diagonal is 16 cm. What is the length of the other diagonal? Find the area?
Answer:
Perimeter = 40 cm
One side = 10 cm
∆POQ is a right-angled triangle
Since d₁ = 16 cm, OP = 8 cm
8² + OQ² = 10²
OQ² = 100 – 64 = 36

OQ = 6 cm
QS = 12 cm
d₁ = 16, d₂ = 12 cm
Area = \(\frac {1}{2}\) × d₁ × d₂
= \(\frac {1}{2}\) × 16 × 12
= 8 × 12
= 96 sq.cm

Question 37.
What is the maximum area of a parallelogram with sides 8 cm and 5 cm? What is the speciality of the parallelogram of maximum area?
Answer:
The area will be maximum for a rectangle, and the maximum area is 40 cm².

Question 38.
The area of a rhombus is 112 sq. cm, and one of its diagonals is 16 cm long. Find the length of the other diagonal.
Answer:
Area of the rhombus = \(\frac {1}{2}\) × d₁ × d₂ = 112 sq. cm
⇒ \(\frac {1}{2}\) × d₂ × 16 = 112
⇒ d₂ = 14 cm

Class 8 Maths Chapter 11 Notes Kerala Syllabus Parallelograms

Parallelograms
A parallelogram is a quadrilateral where both pairs of opposite sides are parallel.
In any parallelogram, opposite sides are equal in length.

For example, ABCD is a parallelogram. Therefore, we can say,
AB and CD are parallel and equal
AD and BC are parallel and equal

Diagonals
The diagonals of a parallelogram bisect each other. This means they cut each other into two equal parts.
If the diagonals of a parallelogram are equal in length, the figure is a rectangle.

In the general parallelogram ABCD, we can relate the following properties:

• Two pairs of parallel sides: AB || DC and AD || BC
• Opposite sides are equal: AB = DC and AD = BC
• Equal opposite angles: ∠A = ∠C and ∠B = ∠D
• Diagonals bisect each other: AO = CO and DO = BO

In any parallelogram, the diagonals bisect each other; on the other hand, any quadrilateral in which the diagonals bisect each other is a parallelogram.

Drawing a Parallelogram
1. Sides and angles between them are given
Example:
Draw AB = 6 cm.
Draw lines of length 4 cm, 45° slanted at both ends of AB.
Join the ends of these lines.

2. Diagonals and the angle between them are given
Example:
Draw AC = 6 cm.
Mark the midpoint of AC.
Through the midpoint of AC, draw a line that makes an angle of 70° with the first line.
Mark B and D are on either side of the midpoint, which is 3 cm from it.
Complete the figure.

3. One diagonal and two sides are given.
Example:
Make a triangle with sides 7 cm, 4 cm, and 8 cm
Taking the 8 cm long side as one side, and the other two sides 7 cm and 4 cm, draw another triangle.
Complete the figure.

4. Two diagonals and one side are given.
Example:
Draw AB = 6 cm, mark ‘O’ which is 4 cm away from A and 3 cm away from B.
Complete AOB, extend line AO and BO at C and D respectively, OC = 4 cm and OD = 3 cm.
Complete the figure.

Area of a Parallelogram

The area of a parallelogram is the product of the length of one side and its distance to the opposite side.
Area = base × height = b × h

Rhombuses
A rhombus is a special parallelogram where all four sides are equal.

Area of Rhombuses
The area of any rhombus is half the product of the diagonals

The area of any square is half the square of its diagonal.