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Let the consecutive vertices of a square S be A,B,C &D. Let E,F & G be the mid-points of the sides AB, BC & AD respectively of the square. Then the ratio of the area of the quadrilateral EFDG to that of the square S is nearest to
  • a)
    >(a) 1/2
  • b)
    1/3
  • c)
    1/4
  • d)
    1/8
  • e)
    2/3
Correct answer is option 'A'. Can you explain this answer?
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Let the consecutive vertices of a square S be A,B,C &D. Let E,F & G b...
Quadrilateral EFDG can be split into 2 triangles i.e. GEF and GDF Area of triangle GEF is half the area of AGFB [since all the 4 triangles in AGFB are similar] Area of triangle GDF is half the area of GDCF Hence, the area of quadrilateral EGDF is half the area of square ABCD
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Let the consecutive vertices of a square S be A,B,C &D. Let E,F & G b...
And D, respectively.

To prove that AC and BD are perpendicular, we can use the Pythagorean theorem and the fact that a square has all sides equal in length.

Let's first find the length of the diagonal of the square S. We can use the Pythagorean theorem to do this:

AB^2 + BC^2 = AC^2

Since all sides of a square are equal, we can substitute AB and BC with the length of one side, s:

s^2 + s^2 = AC^2

Simplifying:

2s^2 = AC^2

Taking the square root of both sides:

AC = s√2

Now let's find the length of BD. Since BD is also a side of the square, it has length s.

To prove that AC and BD are perpendicular, we need to show that their slopes multiply to -1. Alternatively, we can use the fact that two lines are perpendicular if their slopes are negative reciprocals of each other.

The slope of AC can be found using the coordinates of points A and C:

slope of AC = (y2 - y1)/(x2 - x1)

Let's assume that point A is at (0, 0) and point C is at (s, s):

slope of AC = (s - 0)/(s - 0) = 1

The slope of BD can be found using the coordinates of points B and D:

slope of BD = (y2 - y1)/(x2 - x1)

Let's assume that point B is at (0, s) and point D is at (s, 0):

slope of BD = (0 - s)/(s - 0) = -1

Since the slopes of AC and BD are negative reciprocals of each other (-1 and 1/1), we can conclude that AC and BD are perpendicular.
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The chief scientist of a major vaccine producing company has the responsibility to keep the formula of the vaccine confidential. He stores the formula in his e-device with 3 consecutive locks, all of which need to be unlocked to access the formula. The first of the 3 locks is a pattern lock, the second one is a numeric lock and the third and final one is an alphanumeric lock. The pattern lock is in the form of a regular hexagon with 6 distinct vertices A1, A2, A3, A4, A5, A6, in that order. One needs to connect all the 6 vertices one after the another in any random order connecting each vertex only once and only one order of connecting the 6 vertices exists which will unlock the first lock. The numeric lock has a code of ABC, where A, B and C are single digit natural numbers. A is a multiple of 2. B is a multiple of 3, C is a multiple of 4 but not equal to A.The alphanumeric lock has a code MNPQRS, where M and N are alphabets and P, Q, R and S are digits. M and N have to be consecutive vowels or consecutive consonants, not necessarily in order. Two alphabets are said to be consecutive vowels if both of them are vowels and they are consecutive when all the 5 vowels are written in alphabetical order. For example, E and I are consecutive vowels, but A and O are not. Two alphabets are said to be consecutive consonants if both of them are consonants and they are consecutive when all the 21 consonants are written in alphabetical order. For example, D and F are consecutive consonants but D and V are not. For example, M and N can be A and E respectively or E and A respectively. P is equal to Base-Ten-Index of M. Q is equal to the Base-Ten-Index of N. R and S can only take binary values, that is, 0 or 1.Base-Ten-Index of an alphabet = Index of alphabet % 10, where Index of an alphabet is its position when all alphabets from A to Z are arranged alphabetically and a%b is the remainder when a is divided by b.For example, Base-Ten-Index of J = 10 = 0 , Base-Ten-Index of M = 13 = 3.The chief scientist can only set passwords/patterns which follow all of the conditions mentioned above.Based on the information given above, answer the questions that follow.Q.Given below is a list of patterns/passwords, some of which are in accordance with the rules mentioned and some are not. The chief scientist can use any combination of valid patterns/passwords from the following to lock the e-device. What is the total number of ways in which he can do so?Pattern:(1) A1 A2 A5 A6 A5 A2 A4 (2) A1 A2 A3 A4 A5 A6 (3) A6 A5 A4 A3 A2 A1 (4) A1 A6 A5 A2 A4Numeric password:(1) 234 (2) 434 (3) 298 (4) 634 (5) 894 (6) 230 (7) 638Alphanumeric password:(1) AE1500 (2) CD3410 (3) HJ8000 (4)QP7600 (5) AB1211 (6) YZ4600 Correct answer is '40'. Can you explain this answer?

The chief scientist of a major vaccine producing company has the responsibility to keep the formula of the vaccine confidential. He stores the formula in his e-device with 3 consecutive locks, all of which need to be unlocked to access the formula. The first of the 3 locks is a pattern lock, the second one is a numeric lock and the third and final one is an alphanumeric lock. The pattern lock is in the form of a regular hexagon with 6 distinct vertices A1, A2, A3, A4, A5, A6, in that order. One needs to connect all the 6 vertices one after the another in any random order connecting each vertex only once and only one order of connecting the 6 vertices exists which will unlock the first lock. The numeric lock has a code of ABC, where A, B and C are single digit natural numbers. A is a multiple of 2. B is a multiple of 3, C is a multiple of 4 but not equal to A.The alphanumeric lock has a code MNPQRS, where M and N are alphabets and P, Q, R and S are digits. M and N have to be consecutive vowels or consecutive consonants, not necessarily in order. Two alphabets are said to be consecutive vowels if both of them are vowels and they are consecutive when all the 5 vowels are written in alphabetical order. For example, E and I are consecutive vowels, but A and O are not. Two alphabets are said to be consecutive consonants if both of them are consonants and they are consecutive when all the 21 consonants are written in alphabetical order. For example, D and F are consecutive consonants but D and V are not. For example, M and N can be A and E respectively or E and A respectively. P is equal to Base-Ten-Index of M. Q is equal to the Base-Ten-Index of N. R and S can only take binary values, that is, 0 or 1.Base-Ten-Index of an alphabet = Index of alphabet % 10, where Index of an alphabet is its position when all alphabets from A to Z are arranged alphabetically and a%b is the remainder when a is divided by b.For example, Base-Ten-Index of J = 10 = 0 , Base-Ten-Index of M = 13 = 3.The chief scientist can only set passwords/patterns which follow all of the conditions mentioned above.Based on the information given above, answer the questions that follow.Q.How many alphanumeric codes for the third lock are possible which necessarily have an A as one of the alphabets in the code?

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Let the consecutive vertices of a square S be A,B,C &D. Let E,F & G be the mid-points of the sides AB, BC & AD respectively of the square. Then the ratio of the area of the quadrilateral EFDG to that of the square S is nearest toa)>(a) 1/2b) 1/3c) 1/4d) 1/8e) 2/3Correct answer is option 'A'. Can you explain this answer?
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Let the consecutive vertices of a square S be A,B,C &D. Let E,F & G be the mid-points of the sides AB, BC & AD respectively of the square. Then the ratio of the area of the quadrilateral EFDG to that of the square S is nearest toa)>(a) 1/2b) 1/3c) 1/4d) 1/8e) 2/3Correct answer is option 'A'. 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 Let the consecutive vertices of a square S be A,B,C &D. Let E,F & G be the mid-points of the sides AB, BC & AD respectively of the square. Then the ratio of the area of the quadrilateral EFDG to that of the square S is nearest toa)>(a) 1/2b) 1/3c) 1/4d) 1/8e) 2/3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let the consecutive vertices of a square S be A,B,C &D. Let E,F & G be the mid-points of the sides AB, BC & AD respectively of the square. Then the ratio of the area of the quadrilateral EFDG to that of the square S is nearest toa)>(a) 1/2b) 1/3c) 1/4d) 1/8e) 2/3Correct answer is option 'A'. Can you explain this answer?.
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