A question along with a set of statements is given below. We have to find which of the given statements is/are sufficient to answer the given question.
Question : Find the area of the triangle PQR.
Statements :
I. PQ = 5 m., PR = 5 m.
II. Length of PO is 4 m.
To find the area of triangle PQR, we examine the given statements.
Statement I: PQ = 5 m, PR = 5 m.
Knowing that PQ = PR = 5 m, we understand that triangle PQR is isosceles with PM as the perpendicular drawn from P to the base RQ.
However, this statement alone doesn't provide information about the base RQ or the height from P to RQ, so it is insufficient to find the area.
Statement II: Length of PO is 4 m.
This statement gives the height PO from the vertex P to the base RQ.
However, without the length of RQ (or the other sides of the triangle), we cannot calculate the area using PO alone.
Combining both statements:
We have PQ = PR = 5 m and PO = 4 m as height.
Triangle POQ is a right triangle with PO as the height and PQ as the hypotenuse. Using the Pythagorean theorem:
\(PQ^2 = PO^2 + OQ^2\)
\(5^2 = 4^2 + OQ^2\)
\(25 = 16 + OQ^2\)
\(OQ^2 = 9\)
\(OQ = 3 \text{ m}\)
Thus, RQ (the base) = 2 × OQ = 6 m.
With base RQ = 6 m and height PO = 4 m, area of triangle PQR is:
\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 4 = 12 \, \text{m}^2\)
Conclusively, both statements together are sufficient to determine the area of the triangle.
Important Questions from Data Sufficiency (Notes)
- In light of the statements I and II choose the most appropriate option.
If a group comprises of five persons A, B, C, D and E, then how many persons are taller than E?
Statement (I) A is taller than B and B is shorter than A and E only.
Statement (II) C is shorter than A and A is shorter than E. - In light of the statements I and II choose the most appropriate option.
Eight boxes- A, B, C, D, P, Q, R and S are stacked vertically but not necessarily in the same order. Which among them is kept immediately above R?
Statement (I) Only three boxes are kept above D and only one box is kept between D and Q. Q is kept lower than D and is immediately below P.
Statement (II) Only one box is kept between A and C. C is kept three boxes above Q. As many boxes are kept above B as are kept below R. 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 select the most appropriate answer.
C, I, L, N, R, and Y live on six different floors of the same building. The lowermost floor in the building is numbered 1, the floor above it is numbered 2, and so on, till the topmost floor is numbered 6. Who among them lives immediately above Y?
(I) Only four people live between N and I. C lives immediately below I.
(II) Only one person lives between C and R. Y lives immediately above N.P and Q are children of S. Who is the father of P? To answer this question which of the statements (1) and (2) is/are necessary?
(1) R is the brother of P and the son of T.
(2) U is the mother of Q.Given below is a question followed by two statements, I and II, each containing some information. Decide which of the statements are sufficient to answer the question.
Question:
Rajesh's salary is Rs. 12,500. What is Arvind's salary?
Statements
I. Arvind gets $\frac{3}{5}$th of the total salary of his and Rajesh'
II. Two-thirds of Arvind's salary is equal to the average salary of both.
