Selina Concise Class 9 Maths Chapter 14 Rectilinear Figures Exercise 14A Solutions
EXERCISE – 14A
(1) The sum of the interior angles of a polygon is four times the sum of it’s a exterior angles. Find the number of sides in the polygon.
Solution :
Sum of interior = 4 × sum of it’s exterior angles.
(2n – 4) × 90 = 4 × 360°
(2n – 4) = 4 × 360°/90
2n – 4 = 4 × 4
2n – 4 = 16
2n = 16 + 4
2n = 20
n = 20/2
n = 10
∴ The number of sides in the polygon is 10.
(2) The angles of the pentagon are in the ratio 4 : 8 : 6 : 4 : 5. Find each of angle of the pentagon.
Solution :
We know that,
Sum of interior angles =
(2n – 4) × 90
= (2 (5) – 4) × 90
= (10 – 4) × 90
= 6 × 90
= 540°
Now,
sum of all interior angles –
4x + 8x + 6x + 4x + 5x = 540
27x = 540
x = 540/27
x = 20
∴ 4x = 4 × 20 = 80°
8x = 8 × 20 = 160°
6x = 6 × 20 = 120°
4x = 4 × 20 = 80°
5x = 5 × 20 = 100°
(3) One angle of a six sided polygon is 140° and the other angles are equal. Find the measure of each equal angle.
Solution :
We know that,
Sum of interior = (2n – 4) × 90 angles of polygon
140 + x + x + x + x + x = [2 × (6) – 4] × 90
140 + 5x = (12 – 4) × 90
140 + 5x = 8 × 90
140 + 5x = 720°
5x = 720° – 140°
5x = 580°
x = 580°/5
x = 116°
∴ The measure of each equal angle is 116°
(4) In a polygon there are 5 right angles and the remaining angles are equal to 195° each. Find the number of sides in the polygon.
Solution :
Sum of all interior angles = (2n – 4) × 90°
5 × 90° + x × 195 = [2 (5 + x) – 4] × 90°
450 + 195x = (10 + 2x – 4) × 90°
450 + 195x = (6 + 2x) × 90°
450 + 195x = 540° + 180°x
195x – 180°x = 540°– 450°
15x = 90°
x = 90°/15
x = 6
Total number of sides = 5 + x
= 5 + 6
= 11 sides
(5) Three angles of a seven-sided polygon are 132° each and the remaining four angles are equal. Find the value of each equal angle.
Solution :
We know that,
Sum of all interior angles = [2n – 4] × 90°
132 + 132 + 132 + = (2 × (7) – 4] × 90°
x + x + x + x = (14 – 4) × 90°
396° + 4x = 10 × 90°
396° + 4x = 900°
4x = 900 – 396°
4x = 504
x = 504/4
x = 126°
∴ The value of each equal angle is 126°
(6) Two angles of an eight-sided polygon are 142° and 176°. If the remaining angles are equal to each other; find the magnitude of each of the equal angles.
Solution :
We know that,
sum of all interior angles of 8 sided polygon = (2n – 4) × 90°
142° + 176° + x + x + x + x + x + x = (2 × (8) – 4) × 90°
142° + 176° + 6x = (16 – 4) × 90°
318 + 6x = 12 × 90°
318 + 6x = 1080
6x = 1080 – 318
6x = 762
x = 762/6
x = 127°
∴ The magnitude of each of the equal angles is 127°.
(7) In a pentagon ABCDE, AB is parallel to DC and ∠A : ∠E : ∠D – 3 : 4 : 5. Find angle E
Solution :
∠B + ∠C = 180
(∵ co-interior angles)
We know that,
sum of all interior angles of pentagon
= [2n – 4] × 90°
3x + 4x + 5x + ∠B + ∠C
= [2 × (5) – 4] × 90°
12x + 180 = [10 – 4] × 90°
12x + 180 = 6 × 90°
12x + 180° = 540°
12x = 540° – 180°
12x = 360°
x = 360°/12
x = 30°
∠E = 4x = 4 × (30°) = 120°
(8) AB, BC and CD are the three consecutive sides of a regular polygon. If ∠BAC = 15°,
Find –
(i) Each interior angle of the polygon.
(ii) Each exterior angle of the polygon
(iii) Number of sides of the polygon.
Solution :
In △ABC,
we know that,
sum of all sides of a triangle = 180°
∠BAC + ∠BCA + ∠ABC = 180°
15° + 15° + ∠ABC = 180°
30° + ∠ABC = 180° – 30°
∠ABC = 150°
(i) Each interior angles is 150°
(ii) Interior angle + Exterior angles = 180°
150° + Exterior angles = 180°
Exterior angles = 180° – 150°
Exterior angle = 30°
(iii) Each exterior angle = 360°/number of sides (n)
30° = 360°/n
n = 360°/30
n = 12
∴ Number of sides of the polygon is 12.
(9) The ratio between an exterior angle and an interior angle of a regular polygon is 2 : 3. Find the number of sides in the polygon.
Solution :
Let exterior angle = 2x
Let interior angle = 3x
Interior angle + Exterior angle = 180°
3x + 2x = 180°
5x = 180°
x = 180°/5
x = 36°
Exterior angle = 2 × (36°) = 72°
∴ Each Exterior angle = 360°/n (number of sides)
72 = 360°/n
n = 360°/72
n = 5
∴ The number of sides in the polygon is 5.\
(10) The difference between an exterior angle of (n – 1) sided regular polygon and an exterior angle of (n + 2) sided regular polygon is 6° find the value of n.
Solution :
Given that, The difference between an exterior angle of (n – 1) sided regular polygon.
∴ Exterior angle = 360°/n – 1
and an exterior angle of (n + 2) sided regular polygon is 6°
∴ Exterior angle = 360°/n + 2
∴ 360/n – 1 – 360°/n + 1 = 6°
360° (1/(n-1) – 1/(n+2) = 6°
360° ((n+2) – (n-1)/(n-1) (n+2)) = 6°
360°/6 ((n+2) – (n-1)/(n-1) (n+2)) = 1
60 (3/n2 + 2n – n – 2) = 1
60 (3/n2 + n – 2) = 1
180 = n2 + n – 2
n2 + n – 2 – 180 = 0
n2 + n – 182 = 0
n2 + 14n – 13n – 182 = 0
n(n + 14) – 13(n + 14) = 0
(n + 14) (n – 13) = 0
n + 14 = 0 or n – 13 = 0
n = – 14 or n = 13
Here is your solution of Selina Concise Class 9 Maths Chapter 14 Rectilinear Figures Exercise 14A
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