Test: Geometry - GMAT MCQ

Statement (1) tells us that all four sides of ABCD are equal. This information alone is not sufficient to determine if ABCD is a rectangle. A square also has all four sides equal, but not all squares are rectangles. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) tells us that all four internal angles of ABCD are equal. If all four angles are equal, it does indicate that ABCD is a rectangle. Rectangles have all interior angles equal to 90 degrees. However, this statement alone does not provide any information about the sides of ABCD, so it is not sufficient to answer the question.

When we consider both statements together, we know that ABCD has equal sides and equal internal angles. This combination is characteristic of a rectangle, as rectangles have both equal sides and equal 90-degree angles. Therefore, both statements together are sufficient to answer the question.

Hence, the correct answer is (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1) tells us that each diagonal of R has a length of 5. However, this information alone is not sufficient to determine the dimensions of R. There are multiple sets of dimensions that could have a diagonal length of 5, so we cannot determine the area based on this statement alone.

Statement (2) tells us that the perimeter of R is 14. The perimeter of a rectangle is given by the formula P = 2 * (length + width). In this case, we have the equation 14 = 2 * (length + width). However, this equation alone is not sufficient to determine the dimensions of R. We have one equation with two variables (length and width), so we need additional information to find the values of length and width individually.

When we consider both statements together, we still do not have enough information to determine the dimensions of R. Statement (1) gives us information about the diagonals, but it does not provide any information about the sides or angles of the rectangle. Statement (2) gives us information about the perimeter, but it does not provide information about the individual side lengths.

Therefore, both statements together are sufficient to answer the question, but neither statement alone is sufficient. The information provided in statements (1) and (2) is not enough to determine the area of rectangular region R.

Hence, the correct answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1) tells us that angle DEF is 96 degrees. Since angle DEF is the vertex angle of the isosceles triangle, it is equal to the sum of the base angles, which are angles EDF and EFD. Therefore, we can conclude that angle EDF is (1/2) * (angle DEF), which is (1/2) * 96 = 48 degrees. Statement (1) alone is sufficient to answer the question.

Statement (2) tells us that angle DFE is 42 degrees. However, this information alone is not sufficient to determine the measure of angle EDF. Without knowing the measure of angle DEF, we cannot determine the measure of angle EDF. Therefore, statement (2) alone is not sufficient to answer the question.

Since statement (1) alone is sufficient to answer the question, but statement (2) alone is not sufficient, the correct answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1) tells us that the pool's top measures 50 feet diagonally. However, this information alone is not sufficient to determine the area of the top. There are multiple sets of dimensions that could have a diagonal of length 50, so we cannot determine the area based on this statement alone.

Statement (2) tells us that one side of the pool's top measures 25 feet. Similarly, this information alone is not sufficient to determine the area of the top. We need information about the other side of the top to calculate the area.

When we consider both statements together, we can determine the dimensions of the pool's top. From statement (1), we know the diagonal length is 50 feet, and from statement (2), we know one side measures 25 feet. This information is sufficient to calculate the dimensions of the rectangle using the Pythagorean theorem. The other side of the top will also be 25 feet, and we can calculate the area by multiplying the length and width.

Therefore, both statements together are sufficient to answer the question asked, but neither statement alone is sufficient. Hence, the correct answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1) tells us that a diagonal line across the front face of K has a length of 40. However, this information alone is not sufficient to determine the dimensions of K. There are multiple sets of dimensions that could have a diagonal of length 40, so we cannot find the volume based on this statement alone.

Statement (2) tells us that a diagonal line across the bottom face of K has a length of 25. Similarly, this information alone is not sufficient to determine the dimensions of K. We cannot find the volume based on this statement alone.

When we consider both statements together, we still do not have enough information to determine the dimensions of K. The two statements provide different diagonal lengths for different faces of the solid, but they do not give us enough information about the relationship between the dimensions.

Therefore, the correct answer is (E) Statements (1) and (2) together are not sufficient to answer the question asked, and additional data is needed.

Statement (1) states that each cookie has a radius of 3 inches. Since the area of a circle is proportional to the square of its radius, all the cookies will have the same area. Therefore, statement (1) alone is sufficient to calculate the number of cookies that can fit in the given metal sheet.

Statement (2) states that five cookies in a row fit exactly within the longer dimension of the sheet. This information provides the number of cookies that can fit in one row but does not provide information about the number of rows. Therefore, statement (2) alone is not sufficient to determine the value of nm.

By considering each statement individually, we see that statement (1) alone is sufficient to answer the question, but statement (2) alone is not sufficient.

Hence, the correct answer is D: EACH statement ALONE is sufficient to answer the question asked.

Statement (1) states that the centers of the three circles lie on a straight line. This information alone does not provide any conclusive evidence regarding whether the circles intersect or not. The alignment of the centers on a straight line does not guarantee that the circles will intersect. For example, if the centers are arranged in a collinear manner, with the circle of radius 2 between the circles of radii 4 and 5, the circles will not intersect. On the other hand, if the centers are arranged such that the circle of radius 2 overlaps with one or both of the other circles, they will intersect. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) states that the greatest distance between the centers of any two circles is 7. This information also does not provide a definitive answer. While the distance between the centers can give us an idea of their relative positions, it does not provide direct information about the intersection of the circles. For example, if the circle of radius 5 is located at the origin, the circle of radius 4 is centered at (7, 0), and the circle of radius 2 is centered at (-3, 0), the circles will not intersect. However, if the circle of radius 2 is centered at (-7, 0), the circles will intersect. Hence, statement (2) alone is not sufficient to answer the question.

By considering both statements together, we still cannot definitively determine if any two circles intersect. While statement (1) provides information about the alignment of the centers, statement (2) only gives the greatest distance between the centers. Without more precise information about the positions and arrangements of the circles, we cannot determine if they intersect.

Therefore, the correct answer is E: Statements (1) and (2) together are NOT sufficient to answer the question asked, and additional data are needed.

Statement (1) states that side LM has a length of 4. However, this statement does not provide any information about the length of side MN or any other side. Therefore, we cannot determine the perimeter of the triangle based on statement (1) alone.

Statement (2) states that side MN has a length of 4√2. Again, this statement does not provide information about the length of any other side of the triangle. Without knowing the length of side LM or any other side, we cannot calculate the perimeter based on statement (2) alone.

By considering both statements together, we still do not have enough information to determine the perimeter. We know the length of one side in statement (1) and the length of another side in statement (2), but we do not know if these sides are equal or if any other side lengths are given. Additional data or the equality of the sides are needed to calculate the perimeter.

Therefore, the correct answer is E: Statements (1) and (2) together are NOT sufficient to answer the question asked, and additional data are needed.

Statement (1) states that z is equal to x + 4. This does not provide any information about the value of y. Since y is not mentioned in this statement, we cannot determine if y is greater than 4 based on statement (1) alone.

Statement (2) states that x is equal to 3 and z is equal to 7. Again, this statement does not provide any information about the value of y. Without any information about y, we cannot determine if y is greater than 4 based on statement (2) alone.

By considering each statement individually, we can see that neither statement alone is sufficient to answer the question about whether y is greater than 4.

Therefore, the correct answer is D: EACH statement alone is sufficient to answer the question asked.

Statement (1): The length of the side of square ABCD is 2.
From this statement alone, we can determine that the side length of square ABCD is 2 units. The area of a square is calculated by squaring the length of its side. Therefore, the area of square ABCD is 2² = 4 square units. This statement alone is sufficient to answer the question.

Statement (2): For square EFGH, which has sides that are 6 longer than those of square ABCD, the ratio of the perimeter to the area is the reciprocal of the corresponding ratio for square ABCD.
This statement provides information about the relationship between the perimeter and area of square EFGH compared to square ABCD. However, it doesn't directly provide any information about the area of square ABCD. Therefore, statement (2) alone is not sufficient to answer the question.

Since statement (1) alone is sufficient to determine the area of square ABCD, but statement (2) alone is not, the correct answer is option D: EACH statement ALONE is sufficient to answer the question asked.