Test: Geometry - GMAT MCQ

To solve this problem, let's analyze the given information:

PS = 10

We need to find the length of segment QR.

Statement (1): PR = 7

This statement provides the length of segment PR, but it does not give any information about segment QR. We cannot determine the length of QR based on this statement alone.

Statement (2): QS = 6

This statement provides the length of segment QS, but it does not give any information about segment QR. We cannot determine the length of QR based on this statement alone.

When we consider both statements together, we have the following information:

PS = 10
PR = 7
QS = 6

However, even with both statements combined, we still cannot determine the length of segment QR. There is no direct or indirect relationship between QR and the given information.

Since neither statement alone nor both statements together are sufficient to determine the length of segment QR, the correct 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: ∠T = 100°
We should recognize that ∠T CANNOT be one of the identical angles. If this were the case, we'd have two angles with measures of 100 degrees each, which would result in a triangle in which the sum of the angles is GREATER than 180 degree (which is IMPOSSIBLE)
So, we can conclude that ∠T must be the LONE angle, which means ∠R and ∠S are the two IDENTICAL angles.
Since the sum of the 3 angles must be 180, we can conclude that ∠R = 40, ∠S = 40, and ∠T = 100
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: ∠S = 40°
Here are two possible cases to consider:
Case a: ∠S is the LONE angle, in which case the ∠R = 70, ∠S = 40, and ∠T = 70
Case b: ∠S is one of the IDENTICAL angles, in which case we could have ∠R = 40, ∠S = 40, and ∠T = 100
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statement (1): The diameter of the unicycle wheel, including the 2-inch tire, was 18 inches.

From this statement, we can calculate the circumference of the wheel. The circumference of a circle is given by the formula C = πd, where C is the circumference and d is the diameter.

In this case, the diameter of the wheel is 18 inches, including the 2-inch tire. So the actual diameter of the wheel (without the tire) is 18 - 2 = 16 inches.

Using the formula for circumference, we can calculate the circumference of the wheel:

C = πd
C = π * 16 inches
C = 16π inches

With this information, we can determine the number of 360-degree rotations made by the wheel while rolling 150 feet.

Statement (2): The wheel traveled 3 inches after it rotated 60°.

This statement alone does not provide sufficient information to calculate the number of rotations or the circumference of the wheel.

However, statement (1) alone is sufficient to determine the circumference of the wheel, and thus, the number of rotations made by the wheel. Therefore, statement (1) alone is sufficient to answer the question.

The correct answer is option D: EACH statement ALONE is sufficient to answer the question asked.

Pre Analysis:
We are given a tree 30 feet high casting a 40 feet shadow 
We are given a shorter tree x feet high 
We are asked the value of x

Statement 1: The shorter tree has a 22 feet shadow
According to this statement, we can make the following measurements in the diagram

• In the above diagram, triangle BCA and DEA are clearly similar and we can say  to get the value of x
• Thus, statement 1 alone is sufficient and we can eliminate options B, C and E

Statement 2: The distance between two trees is 18 feet

• Same as statement 1 
• Thus, statement 2 alone is also sufficient 

Statement (1): MN = NO = OP

This statement tells us that the lengths of three sides of the quadrilateral are equal. However, it does not provide any information about the angles of the quadrilateral. A quadrilateral can have equal side lengths but still not be a square if its angles are not right angles. Therefore, statement (1) alone is not sufficient to determine if MNOP is a square.

Statement (2): Angle N and angle O are right angles

This statement tells us that two angles of the quadrilateral are right angles. However, it does not provide any information about the lengths of the sides of the quadrilateral. A quadrilateral can have right angles but still not be a square if its side lengths are not equal. Therefore, statement (2) alone is not sufficient to determine if MNOP is a square.

When we consider both statements together, we have the following information:

MN = NO = OP (from statement 1)
Angle N and angle O are right angles (from statement 2)

Both conditions for a square are satisfied: equal side lengths and right angles. Therefore, when both statements are considered together, we can conclude that MNOP is a square.

Since both statements together are sufficient to determine that MNOP is a square, but neither statement alone is sufficient, the correct 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 cube has a width of 5 cm.

If the cube has a width of 5 cm, it means all its sides have the same length. Therefore, we know the side length of the cube is 5 cm. With this information, we can calculate the volume of the cube using the formula V = s³, where V is the volume and s is the side length.

Using statement (1) alone, we can determine the volume of the cube.

Statement (2): The cube has a surface area of 150 cm.

The surface area of a cube is given by the formula A = 6s², where A is the surface area and s is the side length.

In this case, we are given the surface area of the cube as 150 cm. However, statement (2) alone does not provide sufficient information to determine the side length of the cube, which is necessary to calculate its volume.

Therefore, statement (1) alone is sufficient to determine the volume of the cube, but statement (2) alone is not sufficient.

The correct answer is option D: EACH statement ALONE is sufficient to answer the question asked.

Statement 1: Each book has a volume of 10 cubic centimeters.
Since we have no information about the size of the drawer, statement 1 is NOT SUFFICIENT

Statement 2: The drawer is 300 centimeters by 30 centimeters by 20 centimeters and has the shape of a rectangular solid.
Since we have no information about the size of the books, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
It's very tempting to think that, since the volume of the drawer = (300)(30)(20) = 180,000 cubic centimeters, we need only divide that this volume by the volume of one book to determine the number of books that will fit in the drawer. To better understand why this reasoning is faulty, let's examine two possible cases:
Case a: The measurements of the book are 1000 cm by 0.1 cm by 0.1 cm (notice that the volume of this book is 10 cubic centimeters). Since this book is far too long to fit into the drawer, the answer to the target question is ZERO books will fit into the drawer
Case b: The measurements of the book are 10 cm by 1 cm by 1 cm (the volume of this book is still 10 cubic centimeters). In this case, the answer to the target question is more than zero books will fit into the drawer
Since we can’t answer the target question with certainty, the combined statements are NOT SUFFICIENT

Volume of cube = s³
Statement 1
√3 ∗ s = √3
We can find s, and get the volume of the cube.
The statement is sufficient.

Statement 2
√2 ∗ s = √2
We can find s, and get the volume of the cube.
The statement is sufficient.
Option D

Statement (1): Side AB measures 5 meters, and Side BC measures 13 meters.

This statement provides the lengths of two sides of the triangle. However, it does not provide information about the angle or the relationship between the sides. Without the information about the height or the relationship between the sides, we cannot determine the area of the triangle based on this statement alone.

Statement (2): Angle ABC measures 90 degrees.

This statement tells us that angle ABC is a right angle. A right angle is a necessary condition for a triangle to be classified as a right triangle. However, this statement alone does not provide any information about the lengths of the sides. Without the side lengths, we cannot determine the area of the triangle based on this statement alone.

When we consider both statements together, we have the following information:

Side AB measures 5 meters, and Side BC measures 13 meters (from statement 1)
Angle ABC measures 90 degrees (from statement 2)

With this combined information, we can conclude that triangle ABC is a right triangle with side lengths of 5, 12, and 13 (a well-known Pythagorean triple). The area of a right triangle can be calculated using the formula A = (1/2) * base * height, where A is the area, base is the length of one side, and height is the length of the other side. In this case, we have the base (5 meters) and the height (12 meters), allowing us to calculate the area.

Therefore, when both statements are considered together, we can determine the area of the right triangle ABC.

Since both statements together are sufficient to answer the question, but neither statement alone is sufficient, the correct 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): Three angles of a quadrilateral are 90 degrees each.

This statement tells us that three angles of the quadrilateral are right angles (90 degrees). However, it does not provide any information about the side lengths of the quadrilateral. A quadrilateral can have three right angles but still not be a square if its side lengths are not equal. Therefore, statement (1) alone is not sufficient to determine if the quadrilateral is a square.

Statement (2): Three sides of a quadrilateral are equal in length.

This statement tells us that three sides of the quadrilateral are equal in length. However, it does not provide any information about the angles of the quadrilateral. A quadrilateral can have equal side lengths but still not be a square if its angles are not right angles. Therefore, statement (2) alone is not sufficient to determine if the quadrilateral is a square.

When we consider both statements together, we have the following information:

Three angles of a quadrilateral are 90 degrees each (from statement 1)
Three sides of a quadrilateral are equal in length (from statement 2)

Both conditions for a square are satisfied: equal side lengths and right angles. Therefore, when both statements are considered together, we can conclude that the quadrilateral is a square.

Since both statements together are sufficient to determine if the quadrilateral is a square, but neither statement alone is sufficient, the correct answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.