Each item contains a Question followed by two Statements. Answer each item using the following instructions :
Choose option
(a) If the Question can be answered by one of the Statements alone, but not by the other.
(b) If the Question can be answered by either Statement alone.
(c) If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.
(d) If the Question cannot be answered even by using both Statements together.
Question : What is the perimeter of the quadrilateral?
Statement-I: \(AB + DC = 10\) cm
Statement-II: \(AD + BC = 10\) cm
Analyzing the Tangential Quadrilateral Properties
The problem involves a quadrilateral \(ABCD\) where a circle touches all its four sides (\(AB\), \(BC\), \(CD\), \(DA\)). Such a quadrilateral is called a tangential quadrilateral.
A key property of a tangential quadrilateral is that the sums of its opposite sides are equal.
For quadrilateral \(ABCD\), this property means:
\(AB + CD = AD + BC\)The question asks for the perimeter of the quadrilateral. The perimeter (\(P\)) is the sum of all its sides:
\(P = AB + BC + CD + DA\)We can rewrite the perimeter formula using the sums of opposite sides:
\(P = (AB + CD) + (AD + BC)\)Evaluating Statement-I
Statement-I states that \(AB + DC = 10\) cm.
Because \(ABCD\) is a tangential quadrilateral, we know that the sum of opposite sides are equal:
\(AB + CD = AD + BC\)Using Statement-I, we know one pair of opposite sides sums to 10 cm:
\(AB + CD = 10 \text{ cm}\)Therefore, due to the property of tangential quadrilaterals, the other pair of opposite sides must also sum to 10 cm:
\(AD + BC = 10 \text{ cm}\)However, Statement-I alone only provides the sum for one pair of opposite sides (\(AB + CD\)). To find the perimeter, we need the sum of *both* pairs, which is \((AB + CD) + (AD + BC)\). While we know \(AB + CD = 10\), we deduced \(AD + BC = 10\) using the tangential property, but the statement itself doesn't directly provide the sum needed for the second half of the perimeter calculation independently.
Thus, Statement-I alone is not sufficient to determine the perimeter.
Evaluating Statement-II
Statement-II states that \(AD + BC = 10\) cm.
Again, because \(ABCD\) is a tangential quadrilateral:
\(AB + CD = AD + BC\)Using Statement-II, we know one pair of opposite sides sums to 10 cm:
\(AD + BC = 10 \text{ cm}\)Therefore, the other pair of opposite sides must also sum to 10 cm:
\(AB + CD = 10 \text{ cm}\)Similar to the analysis of Statement-I, Statement-II alone only gives the sum for one pair of opposite sides (\(AD + BC\)). The perimeter requires the sum of both pairs. While we know \(AD + BC = 10\), we deduced \(AB + CD = 10\) using the tangential property. The statement itself doesn't directly provide the sum needed for the first half of the perimeter calculation independently.
Thus, Statement-II alone is not sufficient to determine the perimeter.
Evaluating Both Statements Together
Let's consider both statements together:
- Statement-I: \(AB + DC = 10\) cm
- Statement-II: \(AD + BC = 10\) cm
Since the quadrilateral is tangential, we know that \(AB + CD = AD + BC\).
Statement-I tells us \(AB + CD = 10\) cm.
Statement-II tells us \(AD + BC = 10\) cm.
These two statements are consistent with the property of tangential quadrilaterals.
Now, let's calculate the perimeter \(P\) using both statements:
\(P = (AB + CD) + (AD + BC)\)Substituting the values from Statement-I and Statement-II:
\(P = (10 \text{ cm}) + (10 \text{ cm})\) \(P = 20 \text{ cm}\)Therefore, by using both statements together, we can successfully determine the perimeter of the quadrilateral.
Conclusion
The question "What is the perimeter of the quadrilateral?" can be answered using both Statement-I and Statement-II together, but neither statement alone is sufficient.
This corresponds to option (c).
Similar Questions
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Question : Is \(P > Q\)?
Statement-I: \(x : y = 2 : 1\)
Statement-II: The angle between the two adjacent sides of the parallelogram is \(60^\circ\). - P, Q, R, S are the mid-points of sides AB, BC, CD, DA respectively of a quadrilateral ABCD.
Question : What is the difference in the area of the quadrilateral ABCD and the area of the quadrilateral PQRS?
Statement-I: Area of the quadrilateral ABCD is 100 square unit.
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Question: The largest of five different integers is 8 and least is 2. What is the average of these integers ?
Statement-I: The sum of all the 5 integers is a multiple of 5
Statement-II: The number of odd integers is odd
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Question:
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Statement-I :
\(n\) is odd.
Statement-II :
\(n\) is even.
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Question:
The product of a natural number N and the number M written by the same digits of N in the reverse order is 252. What is the number N?
Statement-I: N+ M = 33
Statement-II: N > MWhich one of the following is correct in respect of the above Question and the Statements?
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question:
In a triangle ABC, \(\angle A = \angle B-\angle C\). Is angle A acute?
Statement-I: ABC is not an obtuse-angled triangle.
Statement-II: Angle C is acute.Which one of the following is correct in respect of the above Question and the Statements?
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Question:
ABCD is a parallelogram with \(\angle ABC=60^\circ\). If the area of the parallelogram is \(7\sqrt{3}\) square units, then what is the perimeter of the parallelogram?
Statement-I : The lengths of the sides AB and DA are prime numbers.
Statement-II: The lengths of the sides are natural numbers each greater than 1 unit.Which one of the following is correct in respect of the above Question and the Statements?
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question:
AB and CD are chords of a circle intersecting at P. If \(AP \times PB\) = 48 square units, then what is \(CP \times PD\) equal to?
Statement-I : AP = 8 units
Statement-II: CP = 10 units
Which one of the following is correct in respect of the above Question and the Statements?A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question:
In a quadrilateral ABCD, AB = 6 units, BC = 18 units, CD = 6 units, DA = 9 units. What is the length of diagonal BD?
Statement-I: The length of BD is an integer greater than 13.
Statement-II: The length of BD is an even integer.Which one of the following is correct in respect of the above Question and the Statements?
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Question:
ABC is an isosceles triangle with AB = AC = 10 units. If the area of the triangle is 48 square units, then what is the length of the base BC?
Statement-I : The length of BC is an even integer.
Statement-II: The height of the triangle is greater than the length of half of the base.Which one of the following is correct in respect of the above Question and the Statements?
Important Questions from Data Sufficiency
Consider two Statements and a Question:
Statement 1: Priya is 4 ranks below Seema and is 31 st from the bottom.
Statement 2: Ena is 2 ranks above Seema and is 37" from the bottom.
Question: What is Seema’s rank from the top in the class of 40 students ?
Which one of the following is correct in respect of the Statements and the Question ?
Consider two Statements and a Question:
Statement 1: Each of A and D is heavier than each of B, E and F, but none of them is the-heaviest.
Statement 2: A is heavier than D, but is lighter than C.
Question: Who is the heaviest among A,B,C, D and E?
Which one of the following is correct in respect of the Statements and the Question ?
Consider two Statements and a Question :
Statement-1 : The last day of the month is a Wednesday.
Statement-2 : The third Saturday of the month was the seventeenth day.
Question : What day is the fourteenth of the given month ?
Which one of the following is correct in respect of the Statements and the Question ?
Consider the Question and two Statements given below :
Question : Is xan integer?
Statement-1 : x / 3 is not an integer.
Statement-2 : 3 xis an integer.
Which one of the following is correct in respect of the Question and the Statements?
Consider the Question and two Statements given below:
Question : What is the age of Manisha?
Statement-1: Manisha is 24 years younger than her mother.
Statement-2 : 5 years later, the ages of Manisha and her mother will be in the ratio 3 : 5.
Which one of the following is correct in respect of the Question and the Statements?
