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:
In a triangle ABC right angled at B, AC = 20 cm. What is the circum- radius of the triangle?
Statement-I: AB = 12 cm
Statement-II: BC = 16 cm
Right Triangle Circum-Radius Analysis
The problem asks for the circum-radius of a right-angled triangle ABC, where the right angle is at vertex B, and the hypotenuse AC measures 20 cm. We need to determine if the provided Statements I and II are necessary to find the answer.
Circum-Radius Properties in Right Triangles
A key property of right-angled triangles is related to their circumcircle and circum-radius:
- The circumcenter of a right-angled triangle is always located at the midpoint of its hypotenuse.
- The circum-radius (denoted as \(R\)) is the distance from the circumcenter to any vertex. Therefore, the circum-radius is exactly half the length of the hypotenuse.
The formula is expressed as:
\(R = \frac{\text{Hypotenuse}}{2}\)
Question Information Analysis
The question itself provides the following crucial information:
- Triangle ABC is right-angled at B.
- The length of the hypotenuse AC is 20 cm.
Using the property mentioned above, we can directly calculate the circum-radius:
\(R = \frac{AC}{2} = \frac{20 \text{ cm}}{2} = 10 \text{ cm}\)
This calculation demonstrates that the question can be answered using only the information given in the question stem itself.
Statement Sufficiency Evaluation
Let's examine the statements to confirm they are not required:
- Statement-I Alone: Provides AB = 12 cm. While this information, combined with AC = 20 cm and the right angle at B, allows us to find BC using the Pythagorean theorem (\(BC = \sqrt{AC^2 - AB^2} = \sqrt{20^2 - 12^2} = \sqrt{400 - 144} = \sqrt{256} = 16\) cm), it does not provide any new information needed to determine the circum-radius, as the hypotenuse length (AC = 20 cm) was already given in the question. Thus, Statement I alone is not needed.
- Statement-II Alone: Provides BC = 16 cm. Similar to Statement I, this information, along with AC = 20 cm and the right angle at B, allows us to find AB (\(AB = \sqrt{20^2 - 16^2} = \sqrt{400 - 256} = \sqrt{144} = 12\) cm). However, it does not add any information necessary for calculating the circum-radius, which is directly derivable from AC = 20 cm given in the question. Thus, Statement II alone is not needed.
- Both Statements Together: Using both AB = 12 cm and BC = 16 cm together with AC = 20 cm verifies the triangle's dimensions (\(12^2 + 16^2 = 144 + 256 = 400 = 20^2\)). This is consistent but still redundant for finding the circum-radius.
Data Sufficiency Conclusion
Since the hypotenuse length (AC = 20 cm) is explicitly provided in the question itself, and the formula for the circum-radius of a right-angled triangle only requires the hypotenuse length, the question is answerable without relying on either Statement I or Statement II.
Therefore, the correct conclusion is that the question can be answered even without using any of 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. - Question : What is the ratio of the lengths of diagonals of a rhombus?
Statement-I: One diagonal of the rhombus is equal to its side.
Statement-II: The longer diagonal of the rhombus is equal to \(\sqrt{3}\) times its side. - The chord of a circle of radius R touches at a point on the circumference of a concentric circle of radius r. The length of the chord is 24 units.
Question : What are the values of r and R?
Statement-I: r is an integer.
Statement-II: R is an integer. - Let a, b, c and d be positive integers.
Question : Which one of a, b, c, d is closest to the product abcd?
Statement-I: \(a > b > c\)
Statement-II: c is not the smallest. - Let \(mn = k\), where m and n are prime numbers and k is an even number.
Question : What is the value of \(mn - n + 1\)?
Statement-I: \(m > n\)
Statement-II: One of the numbers is 2. - Question : If p is a positive integer, then what is the remainder when $p^n$ is divided by $p + 1$?
Statement-I: n is even.
Statement-II: p is even. Question : Is xy positive?
Statement-I: \(x = \sqrt[3]{-0.19683}\)
Statement-II: $y = \(\sqrt[3]{x} \)- Let a, b and c be the sides of a triangle ABC.
Question : Is the triangle equilateral?
Statement-I: \(a^2 + b^2 + c^2 = (ab + bc + ca)\)
Statement-II: \(3a^2 + 3b^2 + 4c^2 = 2ab + 4bc + 4ca\)
Important Questions from Data Sufficiency
- In this question, a question is followed by two statements numbered (I) and (II). You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and decide the appropriate answer.
Q. Harsh, Ketan, Mayank, Rohit, and Sidharth, each have different height. Who is the shortest?
(I) Mayank is taller than Harsh but shorter than Ketan.
(II) Rohit is taller than Harsh but shorter than Mayank. - In this question, a question is followed by two statements numbered (I) and (II). You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and decide the appropriate answer.
Q. Kunal, Mohit, Rohan, Sandeep, and Tarun - each have different age. Who is the oldest?
(I) Mohit is older than Kunal but younger than Sandeep.
(II) Tarun is younger than Sandeep but older than Mohit. - In this question, a question is followed by two statements numbered (I) and (II). You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and decide the appropriate answer.
Question:
Among 5 trees, L, M, N, T and V each having a different height, which tree is the shortest?
Statements:
(I) N is shorter than only three trees.
(II) N is taller than M but shorter than T. Below is given a pair of questions, which are required to be answered, and a triad of statements. Which group of statements is sufficient to answer both the questions?
Question 1: Who is sitting opposite to Q?
Question 2: Who is sitting to the right of Q?
Statement 1 : P and U are sitting on left and right sides of T, respectively
Statement 2: P, Q, R, S, T and U are sitting around a round table with six chairs
Statement 3: Q is NOT sitting on the immediate left side of R
- 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 \(60^\circ\).
