Steps for construction:
Step 1: Draw the ΔABC with the given measures.
Step 2: Construct the perpendicular bisector of any two sides (AB and BC) and let them meet at S which is the circumcenter.
Step 3: With S as centre and SA = SB = SC as radius draw the circumcircle to passes through A, B and C.


Circum radius = 4.3 cm.

n(A - B) = 32 + x, n(B A) = 5x
n(A \(\cap\) B) = x
From the Venn diagram:
Given n(A) = n(B)
32 + x + x = x + 5x
32 + 2x = 6x
32 = 6x - 2x
32 = 4x
x = \({32\over 4}=8\)
The value of x = 8.

Let the people who like tea be T. 
Let the people who like coffee be C 
By using formula 
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
(i) n(T ∩ C) = n(T) + n(C) – n(T ∪ C) = 35 + 20 – 45 = 55 – 45 = 10 
The number of people who like both coffee and tea = 10.
(ii) The number of people who do not like tea 
n(T) = n(U) – n(T) = 45 – 35 = 10
(iii) The number of people who do not like coffee 
n(C) = n(U) – n(C) = 45 – 20 = 25.

Since the diagonals of a rectangle AC and BD are equal and bisect each other

\(\therefore\) OA=OB
\(\angle OAB=\angle OBA=46^0\)
Each angle of a rectangle measures 90°
\(\angle ABC=90^0\)
\(\angle ABO+\angle OBC=90^0\)
\(46^0+\angle OBC=90^0\)
\(\angle OBC=90^0-46^0\)
\(\therefore\angle OBC=44^0\)

The cost of a chocolate Rs (x +y)
Number of chocolates bought by Amir x + y
Total amount paid by him Rs (x +y) (x +y)
= x²+ xy +xy +y²
= x²+ 2xy+ y²
When x = 10 and y = 5
The total amount paid by him (10)² + 2(10)(5) + (5)²
= 100 + 100 + 25 = Rs 225

Given ∠BAD = 120° and AC bisects ∠BAD
\(\angle{BAC}=\frac{1}{2}\times 120^{0}=60^{0}\)
∠1 = ∠2 = 60°
AD || BC and AC is the transversal
∠2 = ∠4 = 60°
Δ ABC is isosceles triangle    [∴ ∠1 = ∠4 = 60 °]
⇒ AB = BC
Parallelogram ABCD is a rhombus.

\({6\over 7}=0.\overline {857142}\)
\({12\over 13}=0.\overline{923076}\)

The two irrational numbers are  0.866142 ...and 0.903076.......

When you join these points, you see that they lie on a line which is parallel to x-axis.