A Question is given followed by two Statements I and II. Consider the Question and the Statements. Which one of the following is correct in respect of the above Question and the Statements?
Question:
The diagonals of a rhombus ABCD are in the ratio 5:12. Is one of the diagonals equal to side of the rhombus?
Statement-I: The sum of the diagonals = 34 cm.
Statement-II: The length of a side = 13 cm.
Problem Setup: Rhombus Diagonals Ratio
The problem asks whether one of the diagonals of a rhombus is equal to its side, given that the diagonals are in the ratio 5:12. We need to determine if the provided statements are necessary to answer this question.
Rhombus Properties and Diagonal Relationships
Key properties of a rhombus relevant to this problem include:
- All four sides are equal in length.
- The diagonals bisect each other at right angles (90 degrees).
Consider a rhombus ABCD where the diagonals AC and BD intersect at point O. The intersection forms four congruent right-angled triangles (e.g., triangle AOB). The sides of triangle AOB are half the lengths of the diagonals (AO = AC/2, BO = BD/2) and the side of the rhombus (AB).
Let the lengths of the diagonals be \(d_1\) and \(d_2\). According to the Pythagorean theorem applied to triangle AOB:
\(s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2\)where s is the length of the side of the rhombus.
Diagonal Ratio Calculation
The question states that the diagonals are in the ratio 5:12. We can represent the lengths of the diagonals as \(d_1 = 5x\) and \(d_2 = 12x\), where x is a positive constant.
Now, let's substitute these into the Pythagorean theorem to find the side length s in terms of x:
\(s^2 = \left(\frac{5x}{2}\right)^2 + \left(\frac{12x}{2}\right)^2\) \(s^2 = \left(\frac{5x}{2}\right)^2 + (6x)^2\) \(s^2 = \frac{25x^2}{4} + 36x^2\)To add these terms, we find a common denominator:
\(s^2 = \frac{25x^2}{4} + \frac{144x^2}{4}\) \(s^2 = \frac{25x^2 + 144x^2}{4}\) \(s^2 = \frac{169x^2}{4}\)Taking the square root of both sides to find s:
\(s = \sqrt{\frac{169x^2}{4}}\) \(s = \frac{13x}{2}\)Diagonal vs. Side Equality Check
The question asks if one of the diagonals is equal to the side of the rhombus. We need to check if \(d_1 = s\) or \(d_2 = s\).
Case 1: Is \(d_1 = s\)?
Check if \(5x = \frac{13x}{2}\).
Multiplying both sides by 2 gives \(10x = 13x\).
Subtracting \(10x\) from both sides gives \(0 = 3x\), which implies \(x = 0\). A value of \(x=0\) would mean the diagonals have zero length, resulting in a degenerate rhombus (a point), which is not typically considered. For any actual rhombus (\(x > 0\)), \(d_1 \neq s\).
Case 2: Is \(d_2 = s\)?
Check if \(12x = \frac{13x}{2}\).
Multiplying both sides by 2 gives \(24x = 13x\).
Subtracting \(13x\) from both sides gives \(11x = 0\), which implies \(x = 0\). Again, this suggests a degenerate rhombus. For any actual rhombus (\(x > 0\)), \(d_2 \neq s\).
Therefore, based purely on the ratio of the diagonals being 5:12, we can definitively conclude that neither diagonal can be equal to the side of the rhombus.
Sufficiency of Question Information
Since the information provided within the question itself (the ratio of the diagonals being 5:12) is sufficient to answer the question "Is one of the diagonals equal to the side of the rhombus?", we do not need to use either Statement I or Statement II.
The analysis above demonstrates that the ratio 5:12 inherently prevents a diagonal from equaling the side length. This means the question can be answered independently of the additional information given in the statements.
Similar Questions
- Area of a rectangle with length x and breadth y is P and area of a parallelogram (which is strictly not a rectangle) with adjacent sides of length x and y is Q.
Question : Is \(P > Q\)?
Statement-I: \(x : y = 2 : 1\)
Statement-II: The angle between the two adjacent sides of the parallelogram is \(60^\circ\). - A circle touches all the four sides AB, BC, CD, DA of a quadrilateral ABCD.
Question : What is the perimeter of the quadrilateral?
Statement-I: \(AB + DC = 10\) cm
Statement-II: \(AD + BC = 10\) cm - 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.
Statement-II: Area of the quadrilateral PQRS is 50 square unit. - A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option.
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
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:
What is the remainder when \(x^{2n}-y^{2n} + 1\) is divided by \(x^n + y^n\), where \(n\) is a natural number?
Statement-I :
\(n\) is odd.
Statement-II :
\(n\) is even.
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:
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?
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
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?
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?
