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Directions: In a square of side 10 m, an isosceles triangle is placed on one of the sides of the square. The base of the triangle is half of the side, while it has the maximum possible height. The triangle creates few portions in the square. A smaller square is placed in the smallest portion with one vertex touching the triangle. The smaller portion now has three compartments because of the square.
What is the ratio of area of the three compartments of the smaller portion (smaller to larger area)?
- a)2:5:8
- b)1:8:1
- c)1:8:16
- d)1:3:7
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Directions: In a square of side 10 m, an isosceles triangle is placed...
Looking at the conditions given in the question we can analyze three phases of the same. Firstly, we focus on the original square and the isosceles triangle inscribed inside it. Second we focus on the square which is inscribed in the smallest portion and the three compartments it creates. Thirdly, the entire focus is on the equilateral triangle and visualizing the distance required to find.
Let x be the length of the square as shown above. We can clearly see, that triangle FGH and triangle AHI are similar to each other. The lines FD and HI are parallel to each other and by AA similarity we can prove it.
FG = 2.5-x, Hl=x, GH = x, AI = 10-x
(2.5-x)(10-x)=x2
25-12.5x+x2 = x2
12.5x=25
X=2
Let the vertex of triangle inside the square be 0.
Length of the side of square = length of the side of equilateral triangle In the figure three we see that OH is the required length
OH2 = OJ2 + HJ2
HJ = 1/2 x = 1 m
OJ = x- height of equilateral triangle
Height of equilateral triangle = 
OJ = 2-√3
OH2 = (2-√3)2 + 12
Ratio to be found using (i), (ii) and (iii) above
0.5:4:8 ⇒ 1:8:16
Hence, the correct option is (c).
The length of the square is two, then in the figure 2 above,
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Directions: In a square of side 10 m, an isosceles triangle is placed...
Understanding the Problem:
In this problem, we have a square with a side of 10m. An isosceles triangle is placed on one of the sides of the square. The base of the triangle is half of the side of the square, with the maximum possible height. A smaller square is placed in the smallest portion created by the triangle.
Calculating Areas:
- The area of the square = (side)^2 = 10 * 10 = 100 sq. m
- The base of the triangle = 10/2 = 5 m
- The height of the triangle can be calculated using Pythagorean theorem, h = sqrt(10^2 - 5^2) = sqrt(75) = 5√3
- Area of the triangle = (1/2) * base * height = (1/2) * 5 * 5√3 = 12.5√3 sq. m
- The area of the smallest portion = Area of the triangle - Area of the inscribed square = 12.5√3 - 25 sq. m = 12.5(√3 - 2) sq. m
Calculating Ratios:
- The area of the smallest portion is divided into three compartments by the inscribed square.
- Let the areas of these compartments be x, y, and z
- x + y + z = 12.5(√3 - 2) sq. m
- As the smaller square is inscribed in the smallest portion, the ratios of the areas of the three compartments are in the ratio of the squares of their sides.
- Let the side of the inscribed square be 's'
- The ratios of the areas are s^2 : (5s)^2 : (10 - 5s)^2
- Simplifying, we get s^2 : 25s^2 : 100 - 100s + 25s^2 = 1 : 25 : 100 - 100s + 25
- Since the sum of the ratios is 1 + 25 + 100 - 100s + 25 = 151 - 100s
- The ratio of the three compartments is 1 : 25 : 100 - 100s
Final Answer:
The ratio of the three compartments of the smaller portion is 1 : 25 : 100 - 100s, which is equivalent to 1 : 3 : 7. Hence, the correct answer is option D.
In this problem, we have a square with a side of 10m. An isosceles triangle is placed on one of the sides of the square. The base of the triangle is half of the side of the square, with the maximum possible height. A smaller square is placed in the smallest portion created by the triangle.
Calculating Areas:
- The area of the square = (side)^2 = 10 * 10 = 100 sq. m
- The base of the triangle = 10/2 = 5 m
- The height of the triangle can be calculated using Pythagorean theorem, h = sqrt(10^2 - 5^2) = sqrt(75) = 5√3
- Area of the triangle = (1/2) * base * height = (1/2) * 5 * 5√3 = 12.5√3 sq. m
- The area of the smallest portion = Area of the triangle - Area of the inscribed square = 12.5√3 - 25 sq. m = 12.5(√3 - 2) sq. m
Calculating Ratios:
- The area of the smallest portion is divided into three compartments by the inscribed square.
- Let the areas of these compartments be x, y, and z
- x + y + z = 12.5(√3 - 2) sq. m
- As the smaller square is inscribed in the smallest portion, the ratios of the areas of the three compartments are in the ratio of the squares of their sides.
- Let the side of the inscribed square be 's'
- The ratios of the areas are s^2 : (5s)^2 : (10 - 5s)^2
- Simplifying, we get s^2 : 25s^2 : 100 - 100s + 25s^2 = 1 : 25 : 100 - 100s + 25
- Since the sum of the ratios is 1 + 25 + 100 - 100s + 25 = 151 - 100s
- The ratio of the three compartments is 1 : 25 : 100 - 100s
Final Answer:
The ratio of the three compartments of the smaller portion is 1 : 25 : 100 - 100s, which is equivalent to 1 : 3 : 7. Hence, the correct answer is option D.
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Directions: In a square of side 10 m, an isosceles triangle is placed on one of the sides of the square. The base of the triangle is half of the side, while it has the maximum possible height. The triangle creates few portions in the square. A smaller square is placed in the smallest portion with one vertex touching the triangle. The smaller portion now has three compartments because of the square.What is the ratio of area of the three compartments of the smaller portion (smaller to larger area)?a)2:5:8b)1:8:1c)1:8:16d)1:3:7Correct answer is option 'D'. Can you explain this answer?
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Directions: In a square of side 10 m, an isosceles triangle is placed on one of the sides of the square. The base of the triangle is half of the side, while it has the maximum possible height. The triangle creates few portions in the square. A smaller square is placed in the smallest portion with one vertex touching the triangle. The smaller portion now has three compartments because of the square.What is the ratio of area of the three compartments of the smaller portion (smaller to larger area)?a)2:5:8b)1:8:1c)1:8:16d)1:3:7Correct answer is option 'D'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Directions: In a square of side 10 m, an isosceles triangle is placed on one of the sides of the square. The base of the triangle is half of the side, while it has the maximum possible height. The triangle creates few portions in the square. A smaller square is placed in the smallest portion with one vertex touching the triangle. The smaller portion now has three compartments because of the square.What is the ratio of area of the three compartments of the smaller portion (smaller to larger area)?a)2:5:8b)1:8:1c)1:8:16d)1:3:7Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Directions: In a square of side 10 m, an isosceles triangle is placed on one of the sides of the square. The base of the triangle is half of the side, while it has the maximum possible height. The triangle creates few portions in the square. A smaller square is placed in the smallest portion with one vertex touching the triangle. The smaller portion now has three compartments because of the square.What is the ratio of area of the three compartments of the smaller portion (smaller to larger area)?a)2:5:8b)1:8:1c)1:8:16d)1:3:7Correct answer is option 'D'. Can you explain this answer?.
Directions: In a square of side 10 m, an isosceles triangle is placed on one of the sides of the square. The base of the triangle is half of the side, while it has the maximum possible height. The triangle creates few portions in the square. A smaller square is placed in the smallest portion with one vertex touching the triangle. The smaller portion now has three compartments because of the square.What is the ratio of area of the three compartments of the smaller portion (smaller to larger area)?a)2:5:8b)1:8:1c)1:8:16d)1:3:7Correct answer is option 'D'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Directions: In a square of side 10 m, an isosceles triangle is placed on one of the sides of the square. The base of the triangle is half of the side, while it has the maximum possible height. The triangle creates few portions in the square. A smaller square is placed in the smallest portion with one vertex touching the triangle. The smaller portion now has three compartments because of the square.What is the ratio of area of the three compartments of the smaller portion (smaller to larger area)?a)2:5:8b)1:8:1c)1:8:16d)1:3:7Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Directions: In a square of side 10 m, an isosceles triangle is placed on one of the sides of the square. The base of the triangle is half of the side, while it has the maximum possible height. The triangle creates few portions in the square. A smaller square is placed in the smallest portion with one vertex touching the triangle. The smaller portion now has three compartments because of the square.What is the ratio of area of the three compartments of the smaller portion (smaller to larger area)?a)2:5:8b)1:8:1c)1:8:16d)1:3:7Correct answer is option 'D'. Can you explain this answer?.
Solutions for Directions: In a square of side 10 m, an isosceles triangle is placed on one of the sides of the square. The base of the triangle is half of the side, while it has the maximum possible height. The triangle creates few portions in the square. A smaller square is placed in the smallest portion with one vertex touching the triangle. The smaller portion now has three compartments because of the square.What is the ratio of area of the three compartments of the smaller portion (smaller to larger area)?a)2:5:8b)1:8:1c)1:8:16d)1:3:7Correct answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
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ample number of questions to practice Directions: In a square of side 10 m, an isosceles triangle is placed on one of the sides of the square. The base of the triangle is half of the side, while it has the maximum possible height. The triangle creates few portions in the square. A smaller square is placed in the smallest portion with one vertex touching the triangle. The smaller portion now has three compartments because of the square.What is the ratio of area of the three compartments of the smaller portion (smaller to larger area)?a)2:5:8b)1:8:1c)1:8:16d)1:3:7Correct answer is option 'D'. Can you explain this answer? tests, examples and also practice CAT tests.
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