Consider two circles, each having radius 5 cm, touching each other at a point P. A direct tangent QR is drawn touching one circle at a point Q and the other circle at R. Inside the region PQR inscribed by the two circles and the tangent, a square ABCD is inscribed with its base AB on the tangent and the other side touching the two circles at points D and C, respectively.
Find the area of the square ABCD.
Find the area of the square ABCD.
Show Hint
- 100 sq. cm
- 4 sq. cm
- \( 4\sqrt{2} \) sq. cm
- None of the other options is correct
- 40 sq. cm
The Correct Option is A
Solution and Explanation
Since the radius of each circle is 5 cm and the circles are tangent to each other, the distance between their centers is \( 2 \times 5 = 10 \) cm. The square ABCD is inscribed inside the tangent region formed by the circles and the tangent QR.
Step 2: Use the geometry of the square and the circles.
We know that the length of the side of the square is equal to the distance between the points of tangency on the circles. The tangent segment from a point outside a circle to the point of tangency is perpendicular to the radius at the point of tangency.
Step 3: Calculate the side of the square.
The side of the square is equal to the distance between the centers of the two circles, which is 10 cm. Since the square is inscribed within the region between the circles, the area of the square is \( 10^2 = 100 \, \text{sq. cm} \).
Step 4: Conclusion.
The area of the square ABCD is \( 100 \, \text{sq. cm} \). Therefore, the correct answer is (A).
Top Questions on Mensuration
- Read the following scenario and answer the THREE questions that follow.
Brijbhushan, a microfinancier, lends money at the rate of Rs.10 per square meter to small farmers at a village. He charges an annual interest rate of 10%. All the farming plots in that village are rectangular, with areas varying between a minimum of 1000 square meters and a maximum of 10,000 square meters.
This year, Brijbhushan has lent money only to five farmers: Aditya, Binod, Chhuttan, Dabloo and Govind. The perimeter of Chhuttan’s plot is 250 meters, with the length and width being at a ratio of 4:1. Aditya’s plot has an area three times the area of Govind’s plot. The area of Aditya’s plot is also the average of the areas of Govind’s plot and Dabloo’s plot. The plots belonging to Aditya, Binod and Dabloo are of the same width, but of different lengths. Moreover, the length of Binod’s plot is the sum of the lengths of Aditya’s plot and Dabloo’s plot. - There are three rectangular tanks in a building. The length, width and height of the first tank are \( m \) meters each, and the length, width and height of the second tank are \( n \) meters each. However, the length, width and height of the third tank are \( m \) meters, \( n \) meters and 1 meter, respectively. Initially, the first tank is full of water, while the second and the third are empty. When the second and the third tanks are completely filled with water transferred from the first tank, 85,000 liters of water is still left in the first tank.
If both \( m \) and \( n \) are positive integers, what is the value of \( m \)? (1 meter\(^3\) = 1000 liters)
- Let \( a_1 < a_2< \dots <a_n \) be the list of all prime numbers less than 25. Define \( X_i = \frac{b_i}{a_i} \), where \( b_i \) is the sum of all \( a_k \) where \( k \) ranges from 1 to \( n \), \( k \neq i \). Let \( B \) be the set of all integer-valued \( X_i \).
What is the SMALLEST element of \( B \)?
- There are three rectangular tanks. The first tank has length, breadth and height each equal to \(m\) meters, the second tank has length, breadth and height each equal to \(n\) meters, and the third tank has length, breadth and height equal to \(m\) meters, \(n\) meters and \(1\) meter respectively. It is known that \(m\) and \(n\) are integers. Initially, the first tank is full of water. Then the second and third tanks are completely filled using the water from the first tank. After this, the first tank still has 85{,000 liters of water. If \(1 \text{ m}^3 = 1000\) liters, find the value of \(m\).}
- A battle commander arranged his soldiers in three concentric circles. The inner radius is 70 m and the outer radius is 140 m. If the distance between any two adjacent soldiers on each circle is 1 m, how many total soldiers are required?
Questions Asked in XAT exam
- Arrange the following generations from the oldest to the youngest:
1. Generation X
2. Silent Generation
3. Generation Alpha
4. Baby Boomers
5. Millennials
- XAT - 2026
- Current Affairs
Light Chemicals is an industrial paint supplier with presence in three locations: Mumbai, Hyderabad and Bengaluru. The sunburst chart below shows the distribution of the number of employees of different departments of Light Chemicals. There are four departments: Finance, IT, HR and Sales. The employees are deployed in four ranks: junior, mid, senior and executive. The chart shows four levels: location, department, rank and gender (M: male, F: female). At every level, the number of employees at a location/department/rank/gender are proportional to the corresponding area of the region represented in the chart.
Due to some issues with the software, the data on junior female employees have gone missing. Notice that there are junior female employees in Mumbai HR, Sales and IT departments, Hyderabad HR department, and Bengaluru IT and Finance departments. The corresponding missing numbers are marked u, v, w, x, y and z in the diagram, respectively.
It is also known that:
a) Light Chemicals has a total of 210 junior employees.
b) Light Chemicals has a total of 146 employees in the IT department.
c) Light Chemicals has a total of 777 employees in the Hyderabad office.
d) In the Mumbai office, the number of female employees is 55.
- XAT - 2026
- Pie Charts
An investment company, Win Lose, recruit's employees to trade in the share market. For newcomers, they have a one-year probation period. During this period, the employees are given Rs. 1 lakh per month to invest the way they see fit. They are evaluated at the end of every month, using the following criteria:
1. If the total loss in any span of three consecutive months exceeds Rs. 20,000, their services are terminated at the end of that 3-month period,
2. If the total loss in any span of six consecutive months exceeds Rs. 10,000, their services are terminated at the end of that 6-month period.
Further, at the end of the 12-month probation period, if there are losses on their overall investment, their services are terminated.
Ratan, Shri, Tamal and Upanshu started working for Win Lose in January. Ratan was terminated after 4 months, Shri was terminated after 7 months, Tamal was terminated after 10 months, while Upanshu was not terminated even after 12 months. The table below, partially, lists their monthly profits (in Rs. ‘000’) over the 12-month period, where x, y and z are masked information.
Note:
• A negative profit value indicates a loss.
• The value in any cell is an integer.
Illustration: As Upanshu is continuing after March, that means his total profit during January-March (2z +2z +0) ≥
Rs.20,000. Similarly, as he is continuing after June, his total profit during January − June ≥
Rs.10,000, as well as his total profit during April-June ≥ Rs.10,000.
- XAT - 2026
- Logical and Analytical Reasoning Skills
Business schools’ (B schools) curriculums are filled with group assignments and case competitions. Even when students have just joined the B schools, corporate houses try 38 to catch good talent early by promising them internships based on case competitions. These competitions involve solving the problems presented by the organizations, analyzing the challenges they currently face, and presenting solutions in a manner that convinces the organizations’ representatives.
For students who are just joining a B school, the capability to actually solve such problems is quite limited. Because of that, the corporate houses generally are more focused on the presentations made by groups. Hence, the groups that communicate better, most often, win these competitions.
Abirami joins MBS, a B school. As a fresher, she believes she needs to learn a lot about how organizations work and wants to work with others who have joined MBS and have work experience.- XAT - 2026
- Statements and Assumptions
- Selvam Tutorials offers exclusive face-to-face evening classes from Monday to Friday for students who are preparing for various entrance exams. Selvam Dindigul, the owner, operates from his ancestral house, situated in the middle of the city. There are several instructors, but each teaches only one of the following courses: quantitative ability, verbal ability and decision making. Selvam usually recruits fresh graduates, or those instructors who are working with other tutorials and seeking a job change. As Selvam Tutorials has been expanding aggressively over the past few years, managing it alone becomes challenging for Selvam. Hence, he recruits Kanmani Guna, his niece, who has an online MBA with specializations in Accountancy and HR, as a manager. He asks Kanmani to introduce modern HR practices at Selvam Tutorials while managing cost effectively.
- XAT - 2026
- Premises