Edudel, Directorate of Education Govt. of NCT of Delhi Class 8 Mental Maths Question Bank Chapter 3 Understanding Quadrilaterals Questions Solution. In this chapter, there is total 36 math problems. We have given each questions solution step by step.
1.) What is the maximum number of obtuse angles that a quadrilateral can have?
ANSWER:
The maximum number of obtuse angles that a quadrilateral has 3.
2.) What is the sum of all interior angles of a hexagon?
ANSWER:
The sum of all interior angles of a hexagon is 7200
3.) How many non over lapping triangles can we make in a polygon having n sides by joining the vertices?
ANSWER:
Given that, polygon having n sides.
(n-2) non over lapping triangles can we make in a polygon having n sides
4.) If two adjacent angles of a parallelogram are ( − °) and ( + °), then find the ratio of these angles.
ANSWER:
Given that,
Two adjacent angles of a parallelogram are ( − °) and ( + °).
We know, sum of adjacent angles of a parallelogram is 1800
( − °) + ( + °) = 1800
15x + 30° =1800
15x = 1800 – 30°
15x =1500
X = 100
( − °) = 5 x 10 – 5 = 450
( + °) = 10 x 10 + 35 = 1350
The ratio of these angles = 450 / 1350
The ratio of these angles = 1/ 3
5.) The angles of a quadrilateral are in the ratio 1:2:3:4. Find the difference between the smallest and the largest angle.
ANSWER:
Given that,
The angles of a quadrilateral are in the ratio 1:2:3:4
Let, the angles of a quadrilateral are x, 2x, 3x and 4x.
We know, Sum of angles of a quadrilateral = 3600
X + 2x + 3x + 4x = 3600
10x = 3600
X = 360
The angles of a quadrilateral are x, 2x, 3x and 4x.
2x = 2 x 36 = 720
3x = 3 x 36 = 1080
4x = 4 x 36 = 1440
The difference between the smallest (360) and the largest angle (1440)
1440 – 360 = 1080
6.) If PQRS is a ∥gm, then find ∠ − ∠.
ANSWER:
Given, PQRS is a ∥gm.
We know, Opposite angles of a parallelogram are equal.
∠ And ∠ are opposite angles.
∠ = ∠.
∠ − ∠ = 0
7.) What the number is of sides of a regular polygon whose exterior angle measures 72°?
ANSWER:
Given that,
Exterior angle measures 72°
We have to find number is of sides of a regular polygon.
We know,
Number of sides of a regular polygon × measure of each exterior angle = 360o.
Number of sides of a regular polygon = 360o/ measure of each exterior angle
Number of sides of a regular polygon = 360o/72°
Number of sides of a regular polygon = 5
8.) If only one diagonal of a quadrilateral bisects the other, then which type of quadrilateral is it?
ANSWER:
We know,
One diagonal of a quadrilateral bisects the other, then quadrilateral is Kite.
9.) The interior angles of a triangle are in the ratio 3:2:1, then what is the ratio of its exterior angles?
ANSWER:
Given that,
The interior angles of a triangle are in the ratio 3:2:1
Let, the interior angles of a triangle are 3x, 2x and x
We know, Sum of angles of a quadrilateral = 1800
3x + 2x + x = 1800
6x = 1800
X = 300
The interior angles of a triangle are 3x, 2x and x.
3x = 3 x 30 = 900
2x = 2 x 30 = 600
x = 1 x 30 = 300
We know,
Sum of interior angle and exterior angle is 1800
Exterior angle of 900 = 1800 – 900 = 900
Exterior angle of 600 = 1800 – 600 = 1200
Exterior angle of 300 = 1800 – 300 = 1500
The ratio of its exterior angles are 3:4:5
10.) If the area of a square is 289 sq.cm, then find the length of its diagonal.
ANSWER:
Given that,
The area of a square is 289 sq.cm
We have to find the length of its diagonal.
We know,
Area of a square = (Side) 2
289 sq.cm = (Side) 2
(Side) = 17 cm.
Now,
Length of Diagonal = √ x (Side)
Length of Diagonal =17√ cm.
11.) If a square has a diagonal of length 12√ cm, find its area.
ANSWER:
Given that,
Square has a diagonal of length 12√ cm.
We have to find the area of a square.
We know,
Length of Diagonal = √ x (Side)
12√ = √ x (Side)
(Side) = 12√6 cm.
We know,
Area of a square = (Side) 2
Area of a square = (12√6) 2
Area of a square = 144 x 6
Area of a square = 864 sq. cm
12.) How many sides does a regular polygon have if each of its interior angle is 160o?
ANSWER:
We have to find number is of sides of a regular polygon.
Interior angle is 160o
Exterior angle = 180o – 160o= 20o
We know,
Number of sides of a regular polygon × measure of each exterior angle = 360o.
Number of sides of a regular polygon = 360o/ measure of each exterior angle
Number of sides of a regular polygon = 360o/20°
Number of sides of a regular polygon = 18
13.) If the sum of all interior angles of a polygon is 1080o, then how many sides does a polygon have?
ANSWER:
Given that,
The sum of all interior angles of a polygon is 1080o
We have to find sides does a polygon have.
We know,
Sum of interior angles of a polygon of n sides = (n-2) ×180o.
1080o = 180n – 360 o
180n = 1080o + 360 o
n = 1440 o / 180 o
n = 8
Sides does a polygon have is 8.
14.) How many diagonals does a regular hexagon have?
ANSWER:
We know,
Number of diagonals in a n-sided polygon= (−) / 2
Given, regular hexagon hence it has 6 sides. n = 6
Number of diagonals = 6(6−) / 2
Number of diagonals of regular hexagon = 9
15.) What is the measure of each interior angle of a regular polygon having 12 sides?
ANSWER:
We know,
Number of sides of a regular polygon × measure of each exterior angle = 360o.
Measure of each exterior angle = 360o/ Number of sides of a regular polygon
Measure of each exterior angle = 360o/12
Measure of each exterior angle = 30o
We know,
Interior angle + Exterior angle = 180o
Exterior angle = 180o – 30o
Exterior angle =150o
16.) The ratio of each exterior angle to each interior angle of a regular polygon is 2:3. What is the number of sides of the polygon?
ANSWER:
Given that,
The ratio of each exterior angle to each interior angle of a regular polygon is 2:3
We know,
Interior angle + Exterior angle = 180o
2x + 3x = 180o
5x = 180o
X = 36 o
Exterior angle = 2x = 2 x 36 o= 720
We know,
Number of sides of a regular polygon × measure of each exterior angle = 360o.
Number of sides of a regular polygon = 360o/ measure of each exterior angle
Number of sides of a regular polygon = 360o/72°
Number of sides of a regular polygon = 5
17.) The longer side of a parallelogram is 8cm. If the shorter side is / times of the longer side, then what is the perimeter of the parallelogram?
ANSWER:
Given that,
The longer side of a parallelogram is 8cm.
The shorter side is / times of the longer side.
The shorter side of a parallelogram = / x 8 = 6 cm.
Perimeter of the parallelogram = 2 x (longer side + shorter side)
Perimeter of the parallelogram = 2 x (8+6)
Perimeter of the parallelogram = 28 cm.
18.) What is the number of diagonals in a polygon of 12 sides?
ANSWER:
We know,
Number of diagonals in a n-sided polygon= (−) / 2
Given, n = 12
Number of diagonals = 12(12−) / 2
Number of diagonals in a 12 sided polygon = 54
19.) A polygon has 27 diagonals. How many sides does it have?
ANSWER:
We know,
Number of diagonals in a n-sided polygon= (−) / 2
Polygon has 27 diagonals.
27 = (−) / 2
54 = (−)
n2 – 3n -54 = 0
After solving,
n = 9
Sides are 9.
20.) In parallelogram ABCD, ∠ is greater than ∠ by 5o. What is the measure of ∠?
ANSWER:
Given that,
In parallelogram ABCD, ∠ is greater than ∠ by 5o
We know Sum of adjacent angles of parallelogram is 180o
∠ + ∠ = 180o
And,
∠ – ∠ = 5o
On solving,
∠ = 92.5 o
Now,
∠ + ∠ = 180o
∠ = 180o – 92.5 o
∠ =87.5 o
Now, ∠ and ∠ are opposite angles, hence they are equal.
∠=87.5 o
21.) The length of two diagonals of a rectangle are (x+3) cm and (2x-7) cm, find the value of x.
ANSWER:
Given that,
The length of two diagonals of a rectangle are (x+3) cm and (2x-7) cm
We have to find the value of x.
We know,
The diagonals of a rectangle are equal and bisect each other.
(x+3) cm = (2x-7) cm
By rearranging,
3 + 7 = 2x – x
X = 10
22.) The angles of a quadrilateral are in the ratio 1:3:4:4, then what is the sum of two greatest angles of the quadrilateral?
ANSWER:
Given that,
The angles of a quadrilateral are in the ratio 1:3:4:4.
Let, the angles of a quadrilateral are x, 3x, 4x and 4x.
We know, Sum of angles of a quadrilateral = 3600
X + 3x + 4x + 4x = 3600
12x = 3600
X = 300
The angles of a quadrilateral are x, 3x, 4x and 4x.
3x = 3 x 30 = 900
4x = 4 x 30 = 1200
4x = 4 x 30 = 1200
The sum of two greatest angles of the quadrilateral = 1200 + 1200
The sum of two greatest angles of the quadrilateral = 2400
23.) In a regular polygon, each interior angle is thrice the exterior angle. What is the number of sides of a polygon?
ANSWER:
Given,
Each interior angle is thrice the exterior angle.
Let, exterior angle is x then interior angle is 3x.
We know,
Interior angle + Exterior angle = 180o
x + 3x = 180o
4x = 180o
X = 45 o
Exterior angle = x = 45 o
We know,
Number of sides of a regular polygon × measure of each exterior angle = 360o.
Number of sides of a regular polygon = 360o/ measure of each exterior angle
Number of sides of a regular polygon = 360o/45°
Number of sides of a regular polygon = 8
24.) The interior angle of a regular polygon is 100o more than its exterior angle. What is the number of sides of the polygon?
ANSWER:
Given,
The interior angle of a regular polygon is 100o more than its exterior angle
We know,
Interior angle + Exterior angle = 180o
Interior angle – Exterior angle = 100o
On solving,
Interior angle = 140o
Interior angle + Exterior angle = 180o
140o+ Exterior angle = 180o
Exterior angle = 40 o
We know,
Number of sides of a regular polygon × measure of each exterior angle = 360o.
Number of sides of a regular polygon = 360o/ measure of each exterior angle
Number of sides of a regular polygon = 360o/40°
Number of sides of a regular polygon = 9
25.) The interior angle of a regular polygon exceeds its exterior angle by 108o. What is the number of sides of the polygon?
ANSWER:
Given,
The interior angle of a regular polygon exceeds its exterior angle by 108o
We know,
Interior angle + Exterior angle = 180o
Interior angle – Exterior angle = 108o
On solving,
Interior angle = 144o
Interior angle + Exterior angle = 180o
144o+ Exterior angle = 180o
Exterior angle = 36 o
We know,
Number of sides of a regular polygon × measure of each exterior angle = 360o.
Number of sides of a regular polygon = 360o/ measure of each exterior angle
Number of sides of a regular polygon = 360o/36°
Number of sides of a regular polygon = 10
26.) Two adjacent angles of a parallelogram are (2x+30) o and (3x-15) o, what is the value of x?
ANSWER:
Given,
Two adjacent angles of a parallelogram are (2x+30) o and (3x-15) o
We know, Sum of adjacent angles of a parallelogram is (180) o
(2x+30) o+ (3x-15) o= 180 o
5x + 15 = 180o
5x = 165o
X = 33
27.) The lengths of the diagonals of a rhombus are 16 cm and 12 cm. what is the perimeter of the rhombus?
ANSWER:
Given,
The lengths of the diagonals of a rhombus are 16 cm and 12 cm.
We have to find, perimeter of the rhombus.
We know, the diagonals of a rhombus bisect each other at right angle.
We use right angle triangle theorem,
(Side of rhombus) 2 = (16 cm/2)2 + (12 cm / 2) 2
(Side of rhombus) 2 = (8)2 + (6) 2
(Side of rhombus) 2 = 100
(Side of rhombus) = 10
Perimeter of the rhombus = 4 x (Side of rhombus)
Perimeter of the rhombus = 40 cm.
28.) Find the value of x
ANSWER:
We know,
Sum of interior angles of a polygon of n sides= (n-2) ×180o.
Here n= 5 side.
Sum of interior angles of a polygon of 5 sides= 540o.
∠ + 80o. = 180o. Angles in linear pair.
∠ = 100o.
∠ + 60o. = 180o. Angles in linear pair.
∠ = 120o.
100o. + 120o. + 70o. + x + (x – 10) = 540o.
X = 1300
30.) ABCD is a rhombus. If ∠ = 35o, find the value of x.
ANSWER:
Given,
ABCD is a rhombus. If ∠ = 35o
From fig, ∠O = 90o
We know, sum of triangle is 180o
∠AO =180 o – 1250
∠AO = 750
∠AO = x ………………. Same angles
X = 750
31.) Find the value of x.
ANSWER:
We know,
Sum of all exterior angles of a polygon=360o.
500 + 1150 + 900 + x = 360o.
X = 1050
32.) ABCD is a parallelogram, Find the values of x, y and z.
ANSWER:
ABCD is a parallelogram.
∠ + 70o. = 180o. Angles in linear pair.
∠ = 110o
∠ = z = 110o……………… opposite angles are equal of parallelogram.
y = 450 ……………… corresponding angles.
∠A + ∠D = 180o. …………….. Adjacent angle of parallelogram
∠A = 180o– 110o
∠A = 70o
∠A = x + 45o
x = 25 o
y = 450
z = 110o
34.) If PQRS is a rhombus, find x.
ANSWER:
If PQRS is a rhombus
In triangle POS
∠ POS = 900
By right angle theorem,
Ps2 = po2 + OS2
(X+1) 2 = X2 + (X-1) 2
On solving,
x = 4
35.) If ABCD is a rectangle, find x.
ANSWER:
We know,
The diagonals of a rectangle are equal and bisect each other.
2x – 1 = x + 5
By rearranging,
2x – x = 5 + 1
X = 6
36.) ABCD is a parallelogram, BC=BE, find x.
ANSWER:
ABCD is a parallelogram, BC=BE
∠A = ∠ C = 1250 ———– OPPOSITE ANGLES
At point C = angles in linear pair is form.
1250+ x = 1800
x = 700
BC=BE
This is isosceles triangle.
∠BCE =∠BEC = 700
Now, sum of angles of triangle is 1800
700 + 700 + x = 1800
X = 400
Others Chapter Solution:
Lesson 1 |
Rational Numbers |
Lesson 2 |