Test: Geometry - GMAT MCQ

Statement 1: ΔABD has perimeter 15.
Knowing the perimeter of triangle ABD is 15 does not provide enough information to determine the perimeter of the rhombus ABCD. A rhombus has all sides equal in length, but we don't have any specific information about the sides of the rhombus or the relationships between the sides and the triangle's perimeter. Therefore, statement 1 alone is not sufficient.

Statement 2: ΔABC is equilateral.
If triangle ABC is equilateral, it means that all sides of the triangle are equal in length. Since ABC is one of the triangles formed by the sides of the rhombus, it implies that all sides of the rhombus are also equal in length. Therefore, statement 2 alone tells us that the rhombus ABCD is equilateral. An equilateral rhombus has all sides equal in length, so we can determine the perimeter of the rhombus if we know the length of one side. However, statement 2 does not provide us with the length of any specific side. Therefore, statement 2 alone is not sufficient.

By considering both statements together, we know that triangle ABC is equilateral and triangle ABD has a perimeter of 15. This information implies that two sides of the rhombus are equal in length, but we still don't have enough information to determine the length of the other two sides or the overall perimeter of the rhombus. Therefore, even when considering both statements together, the information is not sufficient to determine the perimeter of rhombus ABCD.

As a result, both statements together are sufficient to answer the question asked, but neither statement alone is sufficient. 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: D, E, F are midpoints of the sides of the triangle.
This statement provides information about the location of points D, E, and F, which are midpoints of the sides of triangle ABC. However, it doesn't provide any specific information about the size or proportions of the shaded portion. Therefore, statement 1 alone is not sufficient to determine the percent of the shaded portion.

Statement 2: Triangle ABC is an equilateral triangle.
This statement indicates that triangle ABC is equilateral, meaning all three sides are equal in length, and all three angles are equal. However, it doesn't provide any information about the specific location or size of the shaded portion. Therefore, statement 2 alone is not sufficient to determine the percent of the shaded portion.

Considering each statement alone:
Statement 1 alone does not provide enough information to determine the percent of the shaded portion.

Statement 2 alone does not provide enough information to determine the percent of the shaded portion.

Since neither statement alone is sufficient, the answer is option A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked. Additional information is needed to determine the percent of the shaded portion.

Statement 1: At least one side of rectangle R has an integer length.
This statement tells us that at least one side of rectangle R has an integer length. However, it doesn't provide any information about the other side or the proportions of the rectangle. Therefore, statement 1 alone is not sufficient to determine whether R is a square.

Statement 2: The diagonals of rectangle R have integer lengths.
This statement tells us that the diagonals of rectangle R have integer lengths. For a rectangle, the diagonals are equal in length. However, it doesn't provide any information about the sides of the rectangle or whether they are equal. Therefore, statement 2 alone is not sufficient to determine whether R is a square.

Considering both statements together:
Together, we know that at least one side has an integer length and the diagonals have integer lengths. However, we still don't have enough information to determine whether the sides are equal and therefore whether R is a square. Additional information is needed.

Therefore, the answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient. Additional data is needed to determine whether rectangle R is a square.

Statement 1: Angle ABC and BCD are right angles.
This statement tells us that two adjacent angles of ABCD are right angles. In a rectangle, all angles are right angles. However, this statement alone doesn't provide information about the other angles or sides of the quadrilateral. Therefore, statement 1 alone is not sufficient to determine whether ABCD is rectangular.

Statement 2: Two diagonals have the same length.
This statement tells us that the diagonals of ABCD have the same length. In a rectangle, the diagonals are equal in length. However, this statement alone doesn't provide information about the angles or other sides of the quadrilateral. Therefore, statement 2 alone is not sufficient to determine whether ABCD is rectangular.

Considering both statements together:
Together, we know that two adjacent angles are right angles and the diagonals have the same length. In a rectangle, all angles are right angles, and the diagonals are equal in length. Therefore, when both statements are true, we can conclude that ABCD is a rectangle.

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

Statement 1: The area of the front face of the block is 60 square centimeters.
This statement tells us that the front face of the block has an area of 60 square centimeters. However, it doesn't provide information about the other faces of the block. A rectangular block can have different dimensions for its length, width, and height while still having a front face area of 60 square centimeters. Therefore, statement 1 alone is not sufficient to determine if the block is a cube.

Statement 2: The area of a side of the block is 60 square centimeters.
This statement tells us that one of the sides of the block has an area of 60 square centimeters. However, it doesn't provide information about the other sides or the dimensions of the block. Similar to statement 1, a rectangular block can have different dimensions for its length, width, and height while still having a side area of 60 square centimeters. Therefore, statement 2 alone is not sufficient to determine if the block is a cube.

Considering both statements together:
Both statements tell us that the front face and one of the sides of the block have an area of 60 square centimeters. However, this information is still not sufficient to determine if the block is a cube. We don't have any information about the dimensions of the block or whether the remaining sides are also equal in area. Therefore, when considering both statements together, we still can't determine if the block is a cube.

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

Statement 1: Line segments AC and BD are perpendicular bisectors of each other.
This statement indicates that the diagonals AC and BD intersect at a right angle and divide each other into equal halves. In a rhombus, the diagonals are perpendicular bisectors of each other. Therefore, if statement 1 is true, it guarantees that quadrilateral ABCD is a rhombus.

Statement 2: AB = BC = CD = AD
This statement states that all four sides of quadrilateral ABCD are equal in length. In a rhombus, all sides are equal. If statement 2 is true, it guarantees that quadrilateral ABCD is a rhombus.

Considering each statement alone:
Statement 1 alone guarantees that quadrilateral ABCD is a rhombus, as the condition of perpendicular bisectors of the diagonals is a defining characteristic of a rhombus. Therefore, statement 1 alone is sufficient.

Statement 2 alone guarantees that quadrilateral ABCD is a rhombus, as all sides being equal is a defining characteristic of a rhombus. Therefore, statement 2 alone is sufficient.

Since each statement alone is sufficient to determine that quadrilateral ABCD is a rhombus, the answer is option D: EACH statement ALONE is sufficient to answer the question asked.

Statement 1: R is a point on the circle, and the distance from Q to R is equal to the distance from Q to C.
This statement tells us that the distances from Q to R and Q to C are equal. However, it doesn't provide any direct information about whether Q lies on the circle or not. Without knowing the specific geometric relationship between Q, R, and C, we cannot determine if Q is on the circle based on this statement alone.

Statement 2: S is a point on the circle, and the distance from Q to S is equal to the distance from S to C.
Similar to statement 1, this statement tells us that the distances from Q to S and S to C are equal. Again, it doesn't provide direct information about the position of point Q relative to the circle. Without additional information about the geometric relationship between Q, S, and C, we cannot determine if Q is on the circle based on this statement alone.

Considering both statements together:
Even when considering both statements together, we still don't have enough information to determine if point Q lies on the circle with center C. The given information only relates to the distances between Q, R, S, and C but does not establish the geometric arrangement necessary to determine the position of Q on the circle.

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: The height from point B to the hypotenuse is 120.
Knowing the height from point B to the hypotenuse (the side opposite the right angle) is 120, we have sufficient information to calculate the area of the triangle. The height is the length of the perpendicular line segment from point B to the base AC. Since we have the height and the length of the base, we can calculate the area using the formula: Area = (1/2) * base * height. Therefore, statement 1 alone is sufficient to determine the area of the triangle.

Statement 2: The perimeter of the triangle is 680.
Knowing the perimeter of the triangle does not provide direct information about the lengths of the sides or the height. Without the lengths of the base and height, we cannot calculate the area of the triangle. Therefore, statement 2 alone is not sufficient to determine the area of the triangle.

Considering both statements together:
Statement 1 provides the height of the triangle, and statement 2 provides the perimeter. However, the perimeter does not directly provide the lengths of the sides or the height. Therefore, even when considering both statements together, we still don't have enough information to determine the lengths of the base and height, and thus we cannot calculate the area of the triangle.

As a result, the correct answer is D: EACH statement ALONE is sufficient to answer the question asked.

Statement 1: All four internal angles of ABCD are equal.
If all four internal angles of ABCD are equal, it implies that opposite angles are congruent. In a parallelogram, opposite angles are always congruent. Therefore, statement 1 alone is sufficient to determine that quadrilateral ABCD is a parallelogram.

Statement 2: AC, a diagonal of ABCD, divides ABCD into two congruent triangles.
If AC, a diagonal of ABCD, divides ABCD into two congruent triangles, it suggests that opposite sides and opposite angles are congruent. However, this information alone does not guarantee that the opposite sides are parallel, which is a defining characteristic of a parallelogram. Therefore, statement 2 alone is not sufficient to determine if quadrilateral ABCD is a parallelogram.

By considering both statements together, we have information about the equality of internal angles and the division of the quadrilateral into congruent triangles. This combination of information implies that opposite angles and sides are congruent, which is a defining property of a parallelogram. Therefore, both statements together are sufficient to determine that quadrilateral ABCD is a parallelogram.

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

Statement 1: The area of triangle ABC is equal to the area of triangle ACD.
This statement provides information about the areas of two triangles within the quadrilateral. However, it does not provide direct information about the sides or angles of the quadrilateral. There are various types of quadrilaterals with equal area triangles, such as trapezoids or irregular quadrilaterals. Therefore, statement 1 alone is not sufficient to determine if quadrilateral ABCD is a parallelogram.

Statement 2: AD = DC.
This statement provides information about the lengths of two sides of the quadrilateral. If AD is equal to DC, it suggests that opposite sides of the quadrilateral are congruent. However, this information alone does not guarantee that the opposite sides are parallel, which is a defining characteristic of a parallelogram. Therefore, statement 2 alone is not sufficient to determine if quadrilateral ABCD is a parallelogram.

By considering both statements together, we have information about the areas of two triangles and the equality of a side length. However, this information is still not sufficient to determine if the quadrilateral is a parallelogram. Additional information about the angles or the relationships between the sides and angles would be required to make a definitive conclusion. Therefore, when considering both statements together, they are still not sufficient to determine if quadrilateral ABCD is a parallelogram.

As a result, both statements together are not sufficient to answer the question asked, and additional data are needed. The correct answer is E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.