Hello Ananya

Please find the solution

The length of 3 sides of a convex quadrilateral are 4cm, 8cm, and 16cm. How many positive integer values can the fourth side (in cm) take if no two sides have the same length?

A. 23

B. 19

C. 21

D. 24

Sum of any three sides in a quadrilateral is greater than the fourth side

so 4 + 8 + 16 > n

28 > n .... (1)

also , 4 + 8 + n > 16

n > 4 .... (2)

From (1) and (2)

28 > n > 4

Hence , { 5,6,7,8,....., 27} - { 8 , 16}

21 integral values .

Hi Sir, thank you for this. I did it using another approach. I first divided the quadrilateral into 2 triangles and then used the property of the sum of two sides greater than the third. This way, I got the range of the diagonal of the quadrilateral and then the range of the fourth side. But I am getting a total of 19 values. Attaching my method below.

Can you please point out the mistake in my method?

Here, y need not be an integer .

So maximum integral x = 27 ( when 12 > y > 11) and minimum integral x = 5

5 - 27 , 23 values

Sir, how do we know that A,B,E,D are concyclic points (as mentioned in the solution)?

Angle DAB = 90 , Angle AEB = 90 : Angles subtended by BD are equal so points A,E,B, D are concyclic points.

A regular polygon has interior angles of 168°. How many sides does the polygon have ?

A regular polygon has an interior angle that measures 144° , and a side of which is 12 units long . What is the perimeter of the regular polygon ?

Sum of all interior angles = ( n -2) × 180°

[(n -2) × 180° ]/n = 168°

(n -2 ) × 180 = 168n

12n = 360°

n = 30

Alternate Approach :

Each exterior angle : 180 - 168 = 12°

We know sum of all exterior angles of a polygon is always 360°

So , 360/12 = 30 exterior angles.

Hence number of sides = 30 .

For 2nd problem :

Each exterior angle 180 - 144 = 36

Number of sides = 360/36 = 10

Perimeter of the polygon : 10 × 12 = 120 units

PFA the solution

Hello sir, Please share the solution to this problem -

In a quadrilateral ABCD , AB is parallel to CD. AB= 8 cm , BC = 10 cm ,CD = 16 cm and AD =10 cm. Find the sum of the lengths of the diagonals.

Hello Nikita , PFA the solution .

Hello sir, please share the solution to this problem -

Hi Nikita ,

PFA the solution

In a regular polygon of ‘n’ sides, a circle of radius r is inscribed and another circle of radius R is circumscribed. Which of the following is definitely true?

R/r > 1

R/r < 2

R/r ≤ 2

1 < R/r ≤ 2

for instance take a equilateral triangle r=a/2√3 and R=a/√3

R/r=2.................(1)

now take a square r=a/2 and R=a√2/2

R/r=√2=1.414 ...................(2)

from 1 and 2

1 < R/r ≤ 2

hence option (4)

Three identical squares with bases AP, PQ, and QB are put next to each other to form a rectangle ABCD. Find the sum of the angles <APD + <AQD + <ABD.

60

75

90

120

Hello Sir , kindly share the solution to this problem -

Hi Nikita ,

Please go through this link : https://artofproblemsolving.com/wiki/index.php?title=British_Flag_Theorem

This theorem is valid irrespective of whether P inside/Outside or the boundary of the rectangle .

Thank you sir 🙂 .Got the correct answer using this

PFA the solution:

PFA the solution:

1/4 of (Area of the square ABCD - area of the smaller circle ) = 1/4 of (100 - 39.2) = 15.2 %