GCAT Spatial Visualization Tasks 4 — Questions and Answers
Question 1: A square piece of cardboard is cut diagonally in half to make two triangles. The two triangles are rearranged to form a rectangle (not a square). What are the proportions of the rectangle?
- 1:1
- 1:2 (Correct answer)
- 2:1
- 1:√2
Correct answer: 1:2
Two right isosceles triangles from a square reassemble into a rectangle with length twice the width (ratio 2:1), or equivalently 1:2.
Question 2: Looking at a transparent box from above, you see a 4×3 rectangle. Looking from the front, you see a 4×2 rectangle. What are the dimensions of the box?
- 4×3×2 (Correct answer)
- 3×4×2
- 2×4×3
- 4×2×3
Correct answer: 4×3×2
Top view (4×3) gives length and width; front view (4×2) gives length and height, so the box is 4 long, 3 wide, and 2 tall.
Question 3: A sphere is cut by a single flat plane. What shape is the cross-section always?
- An ellipse
- A circle (Correct answer)
- A parabola
- An oval
Correct answer: A circle
Any flat plane cutting through a sphere always produces a circular cross-section, regardless of angle.
Question 4: You travel 3 miles north, then 4 miles east. How far are you in a straight line from your starting point?
- 7 miles
- 5 miles (Correct answer)
- 6 miles
- 3.5 miles
Correct answer: 5 miles
Using the Pythagorean theorem: √(3² + 4²) = √(9 + 16) = √25 = 5 miles.
Question 5: A square is rotated 45° and placed inside a larger square so that its corners touch the midpoints of the larger square's sides. What fraction of the larger square does the rotated square cover?
- 1/3
- 1/2 (Correct answer)
- 2/3
- 3/4
Correct answer: 1/2
A square rotated 45° with corners at the midpoints of a larger square covers exactly half the larger square's area.
Question 6: A soldier faces East. He turns left 90°, then turns right 180°, then turns left 90°. What direction is he now facing?
- North
- South
- East (Correct answer)
- West
Correct answer: East
Starting East: left 90° = North; right 180° = South; left 90° = East — back to the original direction.
Question 7: How many distinct ways can a standard 2×2 Rubik's cube face be oriented if you can only rotate the entire cube (not individual pieces)?
- 6
- 12
- 24 (Correct answer)
- 48
Correct answer: 24
A cube has 24 possible orientations (6 faces can be on top × 4 rotations per face).
GCAT Spatial Visualization Tasks 4
A square piece of cardboard is cut diagonally in half to make two triangles.
The two triangles are rearranged to form a rectangle (not a square).
What are the proportions of the rectangle?