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Is the perimeter of triangle ABC greater than 6 cm?
(1) One side of triangle ABC measures 3 cm
(2) The lengths of the three sides of triangle ABC are consecutive integers
- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- d)EACH statement ALONE is sufficient.
- e)Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'D'. Can you explain this answer?
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Is the perimeter of triangle ABC greater than 6 cm?(1) One side of tri...
Steps 1 & 2: Understand Question and Draw Inferences
Let’s say the sides of the triangle are a, b, and c.
Let the perimeter of the triangle be X.
Thus, X = a + b + c
We need to find whether X > 6
Since we don’t have any other information, let’s move on to the analysis of the statement 1.
Step 3: Analyse Statement 1
Statement 1 says: One side of triangle ABC measures 3 cm.
Let’s assume a = 3 cm
Now, we know that the sum of two sides of a triangle is always greater than the third side.
So, we can write:
b + c > a
Adding a to both sides of this inequality, we get:
a + b + c > 2a
That is, X > 2a
Substituting a = 3 cm in this inequality, we get:
X > 6 cm
Thus, Statement 1 alone is sufficient to answer the question.
Step 4: Analyse Statement 2
Statement 2 says: The lengths of the three sides of triangle ABC are consecutive integers.
So, the lengths of the sides are n, (n+1), (n+2) cm, where n is a positive integer.
Perimeter X = n + (n+1) + (n+2)
X = 3n + 3
X = 3(n+1)
Now, we know that the sum of two sides of a triangle is always greater than the third side.
This means, we can write:
n + (n+1) > n + 2
2n + 1 > n + 2
n > 1
Adding 1 to both sides of this inequality:
n + 1 > 2
We know that multiplying both sides of an inequality with a positive number doesn’t change the inequality. Therefore, multiplying both sides of the above inequality with 3:
3(n+1) > 6
This means, X > 6
Thus, Statement 2 alone is sufficient to determine if X > 6
Step 5: Analyse Both Statements Together (if needed)
Since we got the answer to the question in Steps 3 and 4, we don’t need to perform this step.
Answer: Option (D)
Most Upvoted Answer
Is the perimeter of triangle ABC greater than 6 cm?(1) One side of tri...
Statement 1: One side of triangle ABC measures 3 cm.
Statement 2: The lengths of the three sides of triangle ABC are consecutive integers.
To determine if the perimeter of triangle ABC is greater than 6 cm, we need to know the lengths of all three sides.
Analysis of Statement 1:
If one side of triangle ABC measures 3 cm, we cannot determine the lengths of the other two sides. It is possible that the other two sides are very short, resulting in a perimeter less than 6 cm. However, it is also possible that the other two sides are longer, resulting in a perimeter greater than 6 cm.
Since statement 1 alone does not provide sufficient information to determine the perimeter of triangle ABC, we can eliminate answer choices A and D.
Analysis of Statement 2:
If the lengths of the three sides of triangle ABC are consecutive integers, we can consider the possibilities:
- If the three sides are 1, 2, 3, then the perimeter is 6 cm, which is not greater than 6 cm.
- If the three sides are 2, 3, 4, then the perimeter is 9 cm, which is greater than 6 cm.
- If the three sides are 3, 4, 5, then the perimeter is 12 cm, which is greater than 6 cm.
- If the three sides are 4, 5, 6, then the perimeter is 15 cm, which is greater than 6 cm.
Therefore, there are cases where the perimeter of triangle ABC is greater than 6 cm and cases where it is not. Statement 2 alone does not provide a definitive answer.
Since neither statement alone is sufficient, we need to consider both statements together.
Analysis of Both Statements Together:
Considering both statements together, we know that one side of triangle ABC measures 3 cm and the lengths of the three sides are consecutive integers. With this information, we can determine that the three sides are 2, 3, 4. The perimeter of triangle ABC is 2 + 3 + 4 = 9 cm, which is greater than 6 cm.
Therefore, both statements together are sufficient to determine that the perimeter of triangle ABC is greater than 6 cm.
The correct answer is option C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement 2: The lengths of the three sides of triangle ABC are consecutive integers.
To determine if the perimeter of triangle ABC is greater than 6 cm, we need to know the lengths of all three sides.
Analysis of Statement 1:
If one side of triangle ABC measures 3 cm, we cannot determine the lengths of the other two sides. It is possible that the other two sides are very short, resulting in a perimeter less than 6 cm. However, it is also possible that the other two sides are longer, resulting in a perimeter greater than 6 cm.
Since statement 1 alone does not provide sufficient information to determine the perimeter of triangle ABC, we can eliminate answer choices A and D.
Analysis of Statement 2:
If the lengths of the three sides of triangle ABC are consecutive integers, we can consider the possibilities:
- If the three sides are 1, 2, 3, then the perimeter is 6 cm, which is not greater than 6 cm.
- If the three sides are 2, 3, 4, then the perimeter is 9 cm, which is greater than 6 cm.
- If the three sides are 3, 4, 5, then the perimeter is 12 cm, which is greater than 6 cm.
- If the three sides are 4, 5, 6, then the perimeter is 15 cm, which is greater than 6 cm.
Therefore, there are cases where the perimeter of triangle ABC is greater than 6 cm and cases where it is not. Statement 2 alone does not provide a definitive answer.
Since neither statement alone is sufficient, we need to consider both statements together.
Analysis of Both Statements Together:
Considering both statements together, we know that one side of triangle ABC measures 3 cm and the lengths of the three sides are consecutive integers. With this information, we can determine that the three sides are 2, 3, 4. The perimeter of triangle ABC is 2 + 3 + 4 = 9 cm, which is greater than 6 cm.
Therefore, both statements together are sufficient to determine that the perimeter of triangle ABC is greater than 6 cm.
The correct answer is option C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
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Is the perimeter of triangle ABC greater than 6 cm?(1) One side of triangle ABC measures 3 cm(2) The lengths of the three sides of triangle ABC are consecutive integersa)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.d)EACH statement ALONE is sufficient.e)Statements (1) and (2) TOGETHER are NOT sufficient.Correct answer is option 'D'. Can you explain this answer?
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