Test: Geometry - GMAT MCQ

To determine the possible lengths of the third side of a triangle given two side lengths, we can apply the triangle inequality theorem. According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

Let's consider the given side lengths:

Side lengths: 3 and 8
The possible range for the third side length would be:
8 - 3 < third side < 8 + 3
5 < third side < 11
Now let's evaluate the answer choices:

I. 5
This falls within the possible range of third side lengths.

II. 8
This is one of the given side lengths and cannot be the length of the third side in a triangle.

III. 11
This falls within the possible range of third side lengths.

Based on the analysis above, the only possible length for the third side is 5. Therefore, the correct answer is option A: ΙI only.

Let's analyze the possible distances between points X and Z based on the given information.

The distance between W and X is 2, the distance between X and Y is 4, and the distance between Y and Z is 9.

Since all the points are on a line, the distance between X and Z can be obtained by adding the distances between W and X, X and Y, and Y and Z.

Distance between X and Z = Distance between W and X + Distance between X and Y + Distance between Y and Z
Distance between X and Z = 2 + 4 + 9
Distance between X and Z = 15

Now, let's evaluate the answer choices:

A: 3
B: 5
C: 7
D: 9
E: 11

The only distance that matches the possible distance between X and Z is 5. Therefore, the correct answer is option B: 5.

To find the smallest possible angle between any two spokes, we need to determine the maximum number of spokes on the wheel.

For a bicycle wheel, the number of spokes is typically even and can be 2, 4, 6, 8, and so on. However, the question states that there are fewer than six spokes. Therefore, the possible numbers of spokes are 2, 4, or 5.

For 2 spokes, the angle between them is 180 degrees.

For 4 spokes, the angle between any two adjacent spokes is 360 degrees divided by 4, which is 90 degrees.

For 5 spokes, the angle between any two adjacent spokes is 360 degrees divided by 5, which is 72 degrees.

Since we want to find the smallest possible angle, we choose the smallest value among the options, which is 72 degrees.

Therefore, the smallest possible angle between any two spokes is 72 degrees. Hence, the answer is E.

The area of a circle is given by the formula A = πr², where A is the area and r is the radius of the circle.

In this case, the area of circle O is given as 16π. We can equate this to the formula to find the radius:

πr² = 16π

Dividing both sides by π, we get:

r² = 16

Taking the square root of both sides, we get:

r = 4

The length of an arc on a circle is given by the formula L = (θ/360) * 2πr, where L is the length of the arc, θ is the central angle in degrees, and r is the radius of the circle.

In this case, the central angle is given as 45 degrees and the radius is 4. Substituting these values into the formula, we get:

L = (45/360) * 2π(4)
L = (1/8) * 2π(4)
L = (1/8) * 8π
L = π

Hence, the length of the arc is π. Therefore, the correct answer is option A.

In a regular hexagon, the diagonal passing through the center divides the hexagon into two congruent equilateral triangles.

The side length of the hexagon is given as 80 centimeters.

In an equilateral triangle, the length of the diagonal (W) is twice the length of the side.

Therefore, W = 2 * 80 = 160 centimeters.

Hence, the correct answer is option D: 160 centimeters.

Let's assume the height of the rectangular box is h cm.

The area of the base is given by length times width, which in this case is 15 cm * 6 cm = 90 cm².

The area of the top is also 90 cm² since the dimensions are the same.

The sum of the areas of the four sides can be calculated by multiplying the perimeter of the base by the height. The perimeter of the base is (15 cm + 6 cm) + (15 cm + 6 cm) = 42 cm.

So, the sum of the areas of the four sides is 42 cm * h cm = 42h cm².

According to the problem statement, the sum of the areas of the base and top is equal to the sum of the areas of the four sides:

90 cm² + 90 cm² = 42h cm²

180 cm² = 42h cm²

Dividing both sides by 42 cm²:

180 cm² / 42 cm² = h

Simplifying:

30/7 = h

Therefore, the height of the box is 30/7 cm.

Hence, the correct answer is option C: 30/7 cm.

Given that Addington and Bordenview are 65 miles apart and Clearwater is 40 miles from Bordenview, we can use the triangle inequality theorem to determine the possible range of distances between Addington and Clearwater.

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In this case, the possible range for the distance from Addington to Clearwater can be determined by considering the lengths of the other two sides of the triangle:

The distance from Addington to Bordenview is 65 miles.
The distance from Bordenview to Clearwater is 40 miles.
For the distance from Addington to Clearwater to be possible, it must satisfy the triangle inequality:

Distance from Addington to Clearwater < Distance from Addington to Bordenview + Distance from Bordenview to Clearwater

Let's evaluate the options:

A: 15
B: 25
C: 35
D: 105
E: 125

Checking the options against the inequality:

A: 15 < 65 + 40 (True)
B: 25 < 65 + 40 (True)
C: 35 < 65 + 40 (True)
D: 105 < 65 + 40 (False)
E: 125 < 65 + 40 (False)

Therefore, the distance from Addington to Clearwater could be 35 miles, as option C suggests.

Hence, the correct answer is option C: 35.

Let's denote the width of the rectangular flower bed as "w" and the length as "l".

According to the given information, the flower bed is 4 times as long as it is wide, so we have the equation: l = 4w.

If the flower bed were 5 feet shorter and 4 feet wider, it would be a square. This means that the new length would be equal to the new width. Let's denote the new width as "w'" and the new length as "l'".
We have the equation: l' = w' and l' = l - 5 and w' = w + 4.

Substituting the expressions for l and w in terms of w into the equation l' = l - 5, we get: 4w - 5 = w + 4.

Simplifying this equation, we find: 3w = 9, which leads to w = 3.

Substituting w = 3 into the equation l = 4w, we find: l = 4(3) = 12.

Therefore, the length of the flower bed is 12 feet.

Hence, the correct answer is option E: 12 feet.

To find the distance traveled by the point on the fan blade, we need to calculate the circumference of the circular path it travels in 15 seconds.

First, let's find the number of revolutions the fan blade makes in 15 seconds:
Number of revolutions = (300 revolutions/minute) * (15 seconds/60 seconds) = 75 revolutions

Next, let's find the circumference of the circular path:
Circumference = 2πr, where r is the distance from the center to the point on the fan blade
Circumference = 2π * 10 = 20π centimeters

Finally, let's calculate the total distance traveled by the point on the fan blade:
Distance traveled = Number of revolutions * Circumference = 75 * 20π = 1500π centimeters

Therefore, the correct answer is option B: 1500π centimeters.