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
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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?
