Find the length of each side of the given regular dodecagon.

Solution:
Hint:

• A quadrilateral is a closed shape and a type of polygon that has four sides, four vertices, and four angles.

Explanation:

• We have been given information in the question to draw a quadrilateral that is not regular.

We have to draw an irregular quadrilateral that are: rectangle, trapezoid, parallelogram, kite, rhombus.

Explanation:

• We have been given four statements in the question from which we have to choose which statement is true.
• In the given four statements we have been given information about the semicircle, concave polygon, triangle and regular polygon.

Step 1 of 1:
Option A:
A semicircle is a polygon.
No this is not true, because polygon only contain straight lines.
Option B:
A concave polygon is regular
It is not necessary that a concave polygon is regular.
So, it is not true
Option C:
A regular polygon is equiangular
Yes this is true, a regular polygon is equiangular and equilateral.
Option D:
Every triangle is regular
This is not true, because mant triangles are not regulat.
Hence, Option C is correct.

Solution:
Hint:

• A nonagon is a polygon with nine sides and nine angles which can be regular, irregular, concave or convex depending upon its sides and interior angles.

Explanation:

• We have been given the length of each side of a nonagon that is 8 in.
• We have to find the perimeter of the given nonagon.

Step 1 of 1:
We have length of each side of a nanagon 8in
Now the perimeter will be
9 × 8in
72in
Hence, Option C is correct.

Solution:
Hint:

• A regular pentagon is a polygon that has 5 sides all of same length and all the angles of the same measure.

Explanation:

• We have been given the two sides of a regular pentagon in the form of expressions that is -  
• 4x + 7 and x + 16
• We have to find the length of a side of the pentagon.

Step 1 of 1:
The length of the two sides of a regular pentagon is represented by x + 16; 4x + 7
Now, We know that all sides of regular polygon are equal.
So,
X + 16 = 4x + 7
3x = 16 - 7
3x = 9
x = 3
And the measure of length will be
= x + 16
= 3 + 16
= 19

Solution:
Hint:

• A polygon whose length of all sides is equal with equal angles at each vertex is called regular polygon.

Explanation:

• We have been given the two sides of a regular polygon that is - (2𝑥 + 27)° 𝑎𝑛𝑑 (3𝑥 − 3)°
• We have to find the value of x and measure of each angle.

Step 1 of 1:
We know that a regulat polygon is equiangular
So,
2x + 27 = 3x - 3
x = 27 + 3
x = 30
And the value of each angle will be
= 2x + 27
= 2(30) + 27
= 87

Hint :- using the additive property and subtraction property ,division property on both sides .solve for x.
Ans:- x = 1
Explanation :-
Given ,8(-x − 6) = -50-10x.
By left distributive property - 8x − 48 = - 50 -10x
Adding 48 on both sides by additive property of equality both sides remain equal.
- 8x − 48 + 48 = - 50 -10x + 48
- 8x = -10x - 2
Adding 10x on both sides by additive property of equality both sides remain equal.
- 8x +10x = -10x - 2 +10x
2x = - 2
Dividing 2   by division property of equality both sides remains equal.

x = -1
∴ x = -1

Solution:
Hint:

• the base angle theorem states that if the sides of a triangle are congruent then the angles opposite these sides are congruent.
• An equilateral triangle is a triangle with all the three sides of equal length.

Explanation:

• We have to find the measure of each angle of an equilateral triangle using base angle theorem.

Step 1 of 1:
Let a triangle be ABC

Here,
AB = AC
Using base angle theorem

And, 
So, 
Therefore,

Step 2 of 2:
We know that the sum of all angles of a triangle is 180⁰.
Now,




So,

Solution:
Hint:

• A regular pentagon is a polygon that has 5 sides all of same length and all the angles of the same measure.

Explanation:

• We have been given in the question the length of each side of a regular pentagon which is (x+5) cm
• We have also been given the perimeter that is 50 cm.
• We have to find the value of x.

Step 1 of 1:
We have given a perimeter of a regular pentaogn 50.
A pentagon has  sides.
The length of the side is  x+5
So,
5(x + 5) = 50
x + 5 = 10
x = 5
Hence, Option A is correct.

Hint :- The reflexive property states that any real number, a, is equal to itself i.e a = a.
Ans :- Option A
Explanation :-
The reflexive property states that any real number, a, is equal to itself. That is, a = a.
Similarly the segment is congruent to itself .
∴Option A

Solution:
Hint:

• the base angle theorem states that if the sides of a triangle are congruent then the angles opposite these sides are congruent.

Explanation:

• We have been given a diagram of a triangle in the question named ABC we have also been given 𝑚∠𝐵 = 55°.
• We have to find out 𝑚∠A.

Step 1 of 1:
We have given figure

Here, AB = AC
It means the given triangle is an isosceles triangle.
Now,
By base angle theorem .
And it is given 
So, 
Step 2 of 2:
We know that the sum of angle of a triangle is 180⁰

Solution:-
We have given that
2f(x) +f(1-x) = x²     - - -- - - - - -(i)
We have to find the value of f(x)
By replacing x by (1-x) in equation (i)we get,
2f(1-x) + f(x) = (1-x)²
2f(1-x) + f(x) = 1 + x² – 2x         - - - - - -(ii)
Multiplying the equation(i) by 2 we get,
4f(x) + 2f(1-x) = 2x²                - - - - - - (iii)
Subtracting equation (ii) from (iii)
3f(x) = x² + 2x -1
So,

Therefore option (b) is correct.

We have given that inverse of the function f(x)

f^-1(x) = (x+5)/3
For solving this let us take

y = f^-1(x)

y = (x+5)/3
Further solving we get,

x = 3y – 5

f(y) = 3y – 5
Therefore,

f(x) = 3x – 5

Solution:
Hint:

• The base angle theorem states that if the sides of a triangle are congruent then the angles opposite these sides are congruent.

Explanation:

• We have been given in the question a diagram of a triangle named ABC and 𝑚∠𝐴 = 60°.
• We have to find the 𝑚∠𝐴 𝑎𝑛𝑑 𝑚∠𝐶.

Step 1 of 1:
In the given figure, AB = AC.
So, ABC is an isosceles triangle.
So, According to base-angle theorem, the angles opposite the congruent sides are congruent.
So,

Step 2 of 2:
Now we know that the sum of angle of triangle is equal to 180⁰.
So,

Since,
So,





Therefore,

We have given the two sets A and B
A= {x, y, z}
B= { p, q, r, s},
For finding the district relations from B to A we have to take the cartesian product of B and A
B×A = {p, q, r, s} × {x, y, z}
= {(p, x) , (p, y) , (p, z) , (q, x) , (q, y), (q, z) , (r, x) , (r, y), (r, z) , (s, x), (s, y), (s, z)}
Therefore there are 12 distinct relations .

We have given the function

f(x) = x + 1

g(x) = 2x – 3
We have to find the value of

i) f(x) + g(x)

ii) f(x) – g(x)

iii) f(g(x))
Therefore,

i) f(x) + g(x) = x + 1 + 2x – 3

= 3x – 2

ii f(x) – g(x) = x + 1 – (2x – 3)

= x + 1 – 2x + 3

= 4 – x

iii) f(g(x)) = f(2x -3)

= (2x – 3) + 1

= 2x – 2
Therefore, f+g = 3x – 2

f – g = 4 – x

f(g) = 2x – 2