If the length of a side of a rhombus is 36 cm and the area of the rhombus is 396 sq. cm, then the absolute value of the difference between the lengths, in cm, of the diagonals of the rhombus is:
Show Hint
Correct Answer: 60
Approach Solution - 1
Let the diagonals of the rhombus be \(d_1\) and \(d_2\). The length of each side of the rhombus is given as \(a = 36\) cm. Step 1: Apply the formula for the area of a rhombus. The area of a rhombus in terms of its diagonals is \[ \text{Area} = \frac{1}{2} d_1 d_2. \] Given that the area is \(396\ \text{cm}^2\), we have \[ \frac{1}{2} d_1 d_2 = 396, \] which gives \[ d_1 d_2 = 792. \] Step 2: Use the relationship between diagonals and side. In a rhombus, the diagonals bisect each other at right angles. Using the Pythagorean theorem, \[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = a^2. \] Multiplying both sides by 4, \[ d_1^2 + d_2^2 = 4a^2 = 4(36^2) = 5184. \] Step 3: Find the difference between the diagonals. Using the identity \[ (d_1 - d_2)^2 = d_1^2 + d_2^2 - 2d_1 d_2, \] and substituting the known values, \[ (d_1 - d_2)^2 = 5184 - 2(792) = 5184 - 1584 = 3600. \] Taking the square root, \[ |d_1 - d_2| = \sqrt{3600} = 60. \] Hence, the required difference between the diagonals is \(60\).
Approach Solution -2
Step 2: Use the diagonal–side relationship. Diagonals of a rhombus bisect each other at right angles, so by Pythagoras: \[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = a^2. \] Multiply by 4: \[ d_1^2 + d_2^2 = 4a^2 = 4(36^2) = 5184. \]
Step 3: Compute \(|d_1 - d_2|\). Use the identity: \[ (d_1 - d_2)^2 = d_1^2 + d_2^2 - 2 d_1 d_2. \] Substitute known values: \[ (d_1 - d_2)^2 = 5184 - 2(792) = 5184 - 1584 = 3600. \] \[ |d_1 - d_2| = \sqrt{3600} = 60. \] Thus, the required difference between the diagonals is: \[ \boxed{60}. \]
Top Questions on Geometry
- In two concentric circles centred at O, a chord AB of the larger circle touches the smaller circle at C. If OA = 3.5 cm, OC = 2.1 cm, then AB is equal to
- In the given figure, PA is tangent to a circle with centre O. If \(\angle APO = 30^\circ\) and OA = 2.5 cm, then OP is equal to
In \(\triangle ABC\), \(DE \parallel BC\). If \(AE = (2x+1)\) cm, \(EC = 4\) cm, \(AD = (x+1)\) cm and \(DB = 3\) cm, then the value of \(x\) is
- For a circle with centre O and radius 5 cm, which of the following statements is true?
P: Distance between every pair of parallel tangents is 10 cm.
Q: Distance between every pair of parallel tangents must be between 5 cm and 10 cm.
R: Distance between every pair of parallel tangents is 5 cm.
S: There does not exist a point outside the circle from where length of tangent is 5 cm. In the adjoining figure, TS is a tangent to a circle with centre O. The value of $2x^\circ$ is
Questions Asked in CAT exam
- The passage given below is followed by four summaries. Choose the option that best captures the essence of the passage.
In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
Cultural appropriation occurs when elements from a particular culture are borrowed without proper understanding, respect, or acknowledgment. This leads to the commodification of sacred symbols, the reinforcement of stereotypes, and the erasure of the cultural context from which these elements originated. It is essential to recognize that the impact of cultural appropriation extends beyond the realm of artistic expression, influencing societal perceptions and perpetuating power imbalances.- CAT - 2025
- Para Summary
- The number of distinct integers $n$ for which $\log_{\left(\frac14\right)}(n^2 - 7n + 11)>0$ is:
- CAT - 2025
- Linear Inequalities
- In the set of consecutive odd numbers $\{1, 3, 5, \ldots, 57\}$, there is a number $k$ such that the sum of all the elements less than $k$ is equal to the sum of all the elements greater than $k$. Then, $k$ equals?
- CAT - 2025
- Number Systems
- The number of distinct pairs of integers $(x, y)$ satisfying the inequalities $x>y \ge 3$ and $x + y<14$ is:
- CAT - 2025
- Number Systems
For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is:
- CAT - 2025
- Linear Inequalities