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?
Circle Geometry: Intersecting Chords Theorem
This question requires understanding a specific property of chords intersecting inside a circle. We are given the product of the segments of one chord and need to find the product of the segments of the other intersecting chord.
Intersecting Chords Theorem Explained
The core concept needed here is the Intersecting Chords Theorem. This theorem is a fundamental result in Euclidean geometry concerning circles. It states that if two chords of a circle intersect internally, then the product of the lengths of the segments on each chord are equal.
Mathematically, if two chords AB and CD intersect at a point P inside a circle, the theorem is expressed as:
\(AP \\times PB = CP \\times PD\)
This equality holds true regardless of the specific lengths of the segments, as long as they form intersecting chords within the same circle.
Applying the Theorem
The question provides the following information:
- AB and CD are chords of a circle intersecting at P.
- The product \(AP \\times PB = 48\) square units.
- We need to determine the value of \(CP \\times PD\).
Using the Intersecting Chords Theorem, we can directly equate the products of the segments:
\(CP \\times PD = AP \\times PB\)
Given that \(AP \\times PB = 48\), we substitute this value into the equation:
\(CP \\times PD = 48\)
Therefore, the value of \(CP \\times PD\) is 48 square units. This answer is derived directly from the information provided in the question statement itself.
Evaluating the Statements
Now, let's assess the role of the two statements provided:
- Statement I: AP = 8 units.
- Statement II: CP = 10 units.
Statement I gives the length of one segment of chord AB. If \(AP = 8\), then \(8 \\times PB = 48\), which means \(PB = 6\). While this tells us the specific lengths of the segments of chord AB, it is not needed to find the value of \(CP \\times PD\), which we already established is 48.
Statement II gives the length of one segment of chord CD. If \(CP = 10\), then \(10 \\times PD = 48\), meaning \(PD = 4.8\). Similar to Statement I, this information provides specific segment lengths for chord CD but is redundant for determining the product \(CP \\times PD\).
Since the value of \(CP \\times PD\) can be determined solely from the initial premise (\(AP \\times PB = 48\)) using the Intersecting Chords Theorem, neither Statement I nor Statement II is necessary.
Final Conclusion on Sufficiency
The question asks for the value of \(CP \\times PD\). The Intersecting Chords Theorem provides a direct relationship: \(CP \\times PD = AP \\times PB\). The value of \(AP \\times PB\) is given as 48 in the question itself. Therefore, the question is answerable using only the information presented in the question, without recourse to either Statement I or Statement II.
This corresponds to the conclusion 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 \(60^\circ\).
