A piece of carpet in the shape of a trapezoid has a smaller base measuring 50 inches and the larger base measuring 70 inches. What is the perimeter of the carpet?
(1) The area of the carpet is 450 square inches.
(2) The height of the carpet is 7.5 inches.

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(E) Statement (1) tells us the area of the piece of carpet. The formula for the area of a trapezoid is:
Area = (0.5(base 1 + base 2) × height).
We can use the given information to solve for the height of the trapezoid:
0.5(70 + 50) × height = 450
60 × height = 450
height = 450/60 = 15/2
height = 7.5.

We cannot determine the perimeter because we know nothing about the lengths of the other two sides. Although you can draw down two perpendiculars to make two right triangles, we don’t know what the base would be for each triangle or if they are the same on both sides of the trapezoid.

In Statement (2), we are given the same information that we solved for in Statement (1). Knowing the height did not allow us to find the perimeter, so it will not help us here either. Statement (2) is insufficient.

Some people might think that you can solve this, but there are an infinite number of possibilities for the lengths of the two sides. Since the statements are both insufficient, and tell us the same thing, combining them will not help us answer the question.

Since the statements are both insufficient, even combined, the correct answer is choice (E).
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I don't get how Statement 1 and Statement 2 are insufficient. Can't you find the length of the two other sides of the trapezoid if the height is found? Could you explain this problem visually?

Let's analyse this. What do we know?
the two bases: 50 and 70.
the height: 7.5

So what trapezoids does it define? We can draw two parallel lines 7.5 inches apart:


The bases lie on this lines. We can fix one of the bases (let's fix the larger one) and move another one to imagine all possible trapezoids that fit the requirements:


We already should understand that the sum of the lengths of sides is not the same for all such trapezoids. But if we do NOT have such understanding and still have doubts, we can take two specific well-known trapezoids:


Draw some heights for easier comprehension:


Then we can easily calculate the sides using Pythagorean theorem:


As wee see, the perimeters in the two cases do differ.

The heights from the top vertices will divide the base in the proposed proportion for a very specific trapezoid (on the right).


However there are infinitely many other trapezoids that satisfy the statements.

The area will stay the same, but not the perimeter.

We looked at the example of such two specific trapezoids above.


The perimeter of the trapezoid on the left is about 149.
The perimeter of the trapezoid on the right is 145.
They are not equal.