Question:
Find the perimeter of a triangle given that:
Statements:
I. Area of the triangle is 24 sq. units.
II. Height of the triangle is 4 units.
Sufficiency Analysis for Triangle Perimeter
The question asks if the given statements are sufficient to determine the perimeter of a triangle.
Statement I Sufficiency
- Statement I says the area of the triangle is 24 sq. units.
- Knowing only the area is insufficient to determine the side lengths and thus the perimeter. Many different triangles can have the same area. For example, a 6x8 right triangle has area 24, and a 4x12 right triangle also has area 24, but their perimeters differ.
Statement II Sufficiency
- Statement II says the height of the triangle is 4 units.
- Knowing only the height is insufficient. The height relates to a specific base, but without knowing the base or other side lengths, the perimeter cannot be found.
Combined Statements Sufficiency
- Combining both statements, we know the area ($A = 24$) and a height ($h = 4$).
- We can find the corresponding base ($b$) using the area formula: $A = \frac{1}{2} \times b \times h$.
- Substituting the values: $24 = \frac{1}{2} \times b \times 4$.
- $24 = 2 \times b$.
- $b = \frac{24}{2} = 12$ units.
- So, we know the triangle has a base of 12 units and the corresponding height is 4 units. However, this information does not uniquely define the triangle.
- Example 1: An isosceles triangle with base 12 and height 4. The equal sides would be $\sqrt{4^2 + (\frac{12}{2})^2} = \sqrt{16 + 36} = \sqrt{52}$. The perimeter would be $12 + 2\sqrt{52} = 12 + 4\sqrt{13}$ units.
- Example 2: A right-angled triangle with legs 4 and 12. The area is $\frac{1}{2} \times 4 \times 12 = 24$. The height corresponding to the base 12 is 4. The perimeter is $4 + 12 + \sqrt{4^2 + 12^2} = 16 + \sqrt{160} = 16 + 4\sqrt{10}$ units.
- Since different possible triangles yield different perimeters ($12 + 4\sqrt{13} \neq 16 + 4\sqrt{10}$), the statements combined are still not sufficient.
Conclusion
Neither statement alone, nor both statements together, provide enough information to uniquely determine the perimeter 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.
